High-school Algebra objective evidence

Read and evaluate an expression — Algebra foundations: expressions, equations and roots

• A fictional bike rental charges $6 once and $4 per hour h. Which expression gives dollars?Automatically checked response; reasoning still matters

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• For the new event n(p)=150−5p on [0,30], with fixed cost 100 dollars and 4 dollars per ticket, construct revenue, cost and profit functions with units. Compare the revenue-maximizing and profit-maximizing prices and their profits.Written reasoning: review required

Fit growth models and change time units — Functions in graphs, measurements and growth decisions

• Rewrite P(t)=20·2^(t/3), t in days, using h in hours and w in weeks. Verify the same prediction at six days.Written reasoning: review required
• Prove equal differences for L(t)=A+mt and equal ratios for E(t)=Cb^t over any interval length h, with C nonzero and b>0. Explain why unequal gaps require adjustment.Written reasoning: review required

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• For the new event n(p)=150−5p on [0,30], with fixed cost 100 dollars and 4 dollars per ticket, construct revenue, cost and profit functions with units. Compare the revenue-maximizing and profit-maximizing prices and their profits.Written reasoning: review required

Read a compound expression as a model — Algebra reasoning, constraints and solution evidence

• For C=45+8n(1+d), n is item count and d a dimensionless surcharge fraction. Name the meanings and units of 8(1+d), n·8(1+d), and the whole expression. Predict the change when n doubles with d fixed.Written reasoning: review required
• For V=P(1+r)^n with P>0,r>−1 and nonnegative integer n, compare doubling P with doubling n. Give a counterexample to the claim both double V.Written reasoning: review required
• A display uses E=(a+b)/(1+u)+c, where a,b,c are energies in joules and u is dimensionless with u>−1. Explain which parts can be treated as entities, then show why E is not a+b/(1+u)+c.Written reasoning: review required

Factor structure and use zero products — Algebra foundations: expressions, equations and roots

• Which is a fully factored form over the integers of 6x² − 15x?Automatically checked response; reasoning still matters
• Solve 2x³ − 18x = 0.Automatically checked response; reasoning still matters
• A learner solves (x − 2)(x + 3) = 6 by setting either factor to zero. Explain the error and solve correctly.Written reasoning: review required

Why every quadratic has two complex roots — Real numbers, complex coordinates, vectors and matrices

• Verify x⁴+16=(x²−2√2 x+4)(x²+2√2 x+4), and factor each quadratic over the complex numbers.Written reasoning: review required

Factor structure and use zero products — Algebra foundations: expressions, equations and roots

• Solve x² − x − 12 = 0.Automatically checked response; reasoning still matters
• The identity x² + kx + 12 = (x + 3)(x + 4) holds for all x. Find k.Automatically checked response; reasoning still matters

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• A proper rectangle has perimeter 28 m and one side x m. Construct its area function and graph description, using factored and vertex forms. State the contextual domain, zero-area interpretation and maximum.Written reasoning: review required

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• For the new event n(p)=150−5p on [0,30], with fixed cost 100 dollars and 4 dollars per ticket, construct revenue, cost and profit functions with units. Compare the revenue-maximizing and profit-maximizing prices and their profits.Written reasoning: review required
• A proper rectangle has perimeter 28 m and one side x m. Construct its area function and graph description, using factored and vertex forms. State the contextual domain, zero-area interpretation and maximum.Written reasoning: review required

Fit growth models and change time units — Functions in graphs, measurements and growth decisions

• An ideal exponential model multiplies by 1.21 per year. What is its equivalent factor per half-year?Automatically checked response; reasoning still matters
• Rewrite P(t)=20·2^(t/3), t in days, using h in hours and w in weeks. Verify the same prediction at six days.Written reasoning: review required
• A quantity grows by 8 percent each month. Give its model from initial value 50, its exact yearly factor and approximate yearly percentage; explain why 96 percent is not exact.Written reasoning: review required

Distinguish repeated addition from repeated multiplication — Algebra II and precalculus: functions and structure

• Derive the finite geometric sum Sₙ=a+ar+…+ar^(n−1) for a positive integer n. State the r=1 case and explain why the formula cannot be divided by 1−r there.Written reasoning: review required
• Find the sum of the first 5 terms with a₁=3 and common ratio 2.Automatically checked response; reasoning still matters
• A fictional savings plan deposits $50 at the end of month 1, then $10 more each month through month 6. Another deposits $50, then 20% more each month through month 6. With no interest, compare their totals exactly. Explain why fitting the first two deposits does not make the plans equivalent.Written reasoning: review required

Explain why expressions are equivalent — Algebra foundations: expressions, equations and roots

• After simplifying 3(2x − 5) − 2(x + 4), what is the coefficient of x?Automatically checked response; reasoning still matters
• What is the constant term in −2(3 − x) + 5x − 1?Automatically checked response; reasoning still matters

Divide polynomials and interpret the remainder — Algebra II and precalculus: functions and structure

• Divide 2x³+3x²−5x+6 by x+2. Give quotient and remainder, then verify by multiplication.Written reasoning: review required

Operate on polynomials and prove identities — Algebra reasoning, constraints and solution evidence

• Let P=2x²−3x+1 and Q=−x²+4. Find P+Q, P−Q and PQ, showing the sign changes and a product coefficient check.Written reasoning: review required
• Explain why sums, differences and products of real polynomials remain real polynomials. Why does this not imply closure under division?Written reasoning: review required

Divide polynomials and interpret the remainder — Algebra II and precalculus: functions and structure

• Find the remainder when P(x)=2x³−3x+5 is divided by x−2.Automatically checked response; reasoning still matters
• For P(x)=x³+kx+6, choose k so that x−2 is a factor.Automatically checked response; reasoning still matters
• Find the remainder when 4x²+3x−2 is divided by 2x−1.Automatically checked response; reasoning still matters
• Explain why the remainder on division by x−a equals P(a), and why P(a)=0 makes x−a a factor. Is the quotient identity P(x)=(x−a)Q(x) valid at x=a when the remainder is zero?Written reasoning: review required

Read roots, multiplicity and end behavior — Algebra II and precalculus: functions and structure

• Enter the distinct real zeros of P(x)=−2(x+1)²(x−3)³.Automatically checked response; reasoning still matters
• For P(x)=(x−1)²(x+2), give the sign on each interval separated by −2 and 1. Explain which root crosses the axis and why.Written reasoning: review required

Construct polynomial and rational graphs — Functions in graphs, measurements and growth decisions

• Construct Q(x)=−(x+2)(x−1)² as a graph or a complete coordinate-and-branch sketch description. Include roots and their multiplicities, crossing/touching behavior, y-intercept, signs and both ends. Supply at least one additional checked landmark.Written reasoning: review required
• Use supplied Graph P and Graph R. A learner says both inputs −1 and 2 are x-intercepts of both graphs because the corresponding factors appear in the formulas. Audit the claim and use the plotted behavior to distinguish the four cases.Written reasoning: review required
• A student draws x⁴−5x²+4 with opposite end directions because it has several roots. Factor it, specify all roots and sign intervals, then repair the ends and explain which algebraic feature controls them.Written reasoning: review required

Why every quadratic has two complex roots — Real numbers, complex coordinates, vectors and matrices

• Verify x⁴+16=(x²−2√2 x+4)(x²+2√2 x+4), and factor each quadratic over the complex numbers.Written reasoning: review required

Classify primitive right triangles — Elementary number theory: divisibility, reciprocity and Diophantine equations

• For m=4, n=1 find the even leg.Automatically checked response; reasoning still matters
• For those parameters find the hypotenuse.Automatically checked response; reasoning still matters
• Derive the parametrization from a primitive triple with odd leg x and even leg y.Written reasoning: review required

Operate on polynomials and prove identities — Algebra reasoning, constraints and solution evidence

• Prove (u²+v²)²=(u²−v²)²+(2uv)². Use u=5,v=2 to construct a right-triangle side triple and check it numerically.Written reasoning: review required
• Prove (a−b)(a+b)=a²−b² and use it to calculate 198·202 without long multiplication. Explain why checking just a=200,b=2 is not the proof.Written reasoning: review required

Read coefficients as choices — Discrete structures: counting, graphs and algorithms

• Find the coefficient of x³ in (1+x)⁶.Automatically checked response; reasoning still matters
• Find the coefficient of x² in (2+x)⁴.Automatically checked response; reasoning still matters
• Which identity is Pascal's recurrence?Automatically checked response; reasoning still matters
• Prove (a+b)ⁿ=Σ(k=0..n) C(n,k)aⁿ⁻ᵏbᵏ for nonnegative integer n.Written reasoning: review required

Divide polynomials and interpret the remainder — Algebra II and precalculus: functions and structure

• Divide 2x³+3x²−5x+6 by x+2. Give quotient and remainder, then verify by multiplication.Written reasoning: review required
• Find the remainder when 4x²+3x−2 is divided by 2x−1.Automatically checked response; reasoning still matters

Separate holes, poles and end behavior — Algebra II and precalculus: functions and structure

• Analyze f(x)=(x²+1)/(x−1): give its domain, slant asymptote and whether it intersects that asymptote. Justify using division.Written reasoning: review required

Solve rational equations without restoring excluded inputs — Algebra foundations: expressions, equations and roots

• Solve 1/x + 1/(x + 1) = 3/2.Automatically checked response; reasoning still matters
• Solve (x + 1)/(x − 3) = 4/(x − 3) + 1. Explain why saying every real number is inaccurate.Written reasoning: review required

Construct division without losing equality — Abstract algebra II: rings, fields and Galois foundations

• Distinguish F[x] from F(x) and prove x has an inverse in only the latter.Written reasoning: review required

Keep domains through rational arithmetic — Algebra reasoning, constraints and solution evidence

• Simplify 2/(x−2)+3/(x+2) and 2/(x−2)−3/(x+2), recording original domains.Written reasoning: review required
• Simplify [(x²−9)/(x²−x)]·[x/(x+3)] and state the complete original domain. Compare it with the simplified formula domain.Written reasoning: review required
• Simplify [(x²−4)/(x−1)] ÷ [(x+2)/(x−1)]. State all restrictions before taking a reciprocal. Why is x=−2 excluded although the final formula is x−2?Written reasoning: review required
• For polynomial expressions p/q and r/s, give formulas for their sum, difference, product and quotient. State evaluation restrictions and explain why the results are rational expressions when the divisor is not the zero rational expression.Written reasoning: review required

Solve and interpret inequalities — Algebra foundations: expressions, equations and roots

• A fictional event has a $17 setup charge and costs $6 per whole ticket. With at most $80, what is the largest nonnegative integer number of tickets?Automatically checked response; reasoning still matters

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• For the new event n(p)=150−5p on [0,30], with fixed cost 100 dollars and 4 dollars per ticket, construct revenue, cost and profit functions with units. Compare the revenue-maximizing and profit-maximizing prices and their profits.Written reasoning: review required

Use exponent laws without losing the base conditions — Algebra II and precalculus: functions and structure

• An original synthetic culture model N(t)=Ab^t has N(1)=30 and N(3)=120, with A>0 and b>0. Find A and b, then predict N(4). State what two observations cannot establish about a real culture.Written reasoning: review required

Construct rational models from rates — Algebra reasoning, constraints and solution evidence

• One pump takes 4 hours longer than another to fill a tank. Together they take 3 hours. Define the unknown, construct an equation and find both individual times.Written reasoning: review required
• A cyclist travels30 km out at v km/hour and 30 km back at(v+5) km/hour in 5 hours total. Construct and solve for v, then compute the whole-trip average speed.Written reasoning: review required
• A machine completes one batch in t hours at a constant rate. Another completes it in 2t hours. A learner says the joint time is 3t. Derive the correct time and explain when the combined-rate model itself could fail.Written reasoning: review required

Find intersections and classify systems — Algebra II and precalculus: functions and structure

• A fictional workshop uses 2 hours and 3 material units for each A, and 1 hour and 4 units for each B. Exactly 14 hours and 32 material units are used. Model and solve for whole nonnegative counts. Explain what the graph intersections mean.Written reasoning: review required

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• A proper rectangle has perimeter 28 m and one side x m. Construct its area function and graph description, using factored and vertex forms. State the contextual domain, zero-area interpretation and maximum.Written reasoning: review required

Solve and interpret inequalities — Algebra foundations: expressions, equations and roots

• A fictional event has a $17 setup charge and costs $6 per whole ticket. With at most $80, what is the largest nonnegative integer number of tickets?Automatically checked response; reasoning still matters

Find intersections and classify systems — Algebra II and precalculus: functions and structure

• A fictional workshop uses 2 hours and 3 material units for each A, and 1 hour and 4 units for each B. Exactly 14 hours and 32 material units are used. Model and solve for whole nonnegative counts. Explain what the graph intersections mean.Written reasoning: review required

Read a linear objective on a polyhedron — Optimization: models, certificates and algorithms

• Explain why replacing an equality by a one-sided inequality can change a resource-allocation model.Written reasoning: review required
• Maximize 3x+2y with x,y≥0, x+y≤4 and x≤2. What is the optimum value?Automatically checked response; reasoning still matters

Build a feasible region from constraints — Algebra reasoning, constraints and solution evidence

• A school makes kits x and posters y, both whole numbers. A kit uses 2 sheets and 3 minutes; a poster uses 1 sheet and 2 minutes. There are 16 sheets and 27 minutes, and at least 3 kits are required. Form the feasible system and decide whether (5,6) and (6,4) are allowed.Written reasoning: review required
• Draw the region x≥0, y≥0, x+y≤5, y>x−1. Label each boundary solid or dashed, give an interior point and decide whether (3,2) belongs.Written reasoning: review required
• Compare x+y≤2 together with x+y>2, and x≥0 together with y≥1. Identify empty or unbounded feasible sets and prove each statement.Written reasoning: review required

Rearrange formulas with conditions — Algebra foundations: expressions, equations and roots

• For nonzero r, which formula isolates t in P = Q(1 + rt), assuming Q is nonzero?Automatically checked response; reasoning still matters
• Solve y = mx + b for x, including m = 0. Explain why x = (y − b)/m is not the whole answer.Written reasoning: review required

Solve equations with reversible steps — Algebra foundations: expressions, equations and roots

• Solve 3(2x − 1) + 4 = 2x + 13.Automatically checked response; reasoning still matters
• Solve (x − 2)/3 + (x + 1)/2 = 4.Automatically checked response; reasoning still matters
• A learner solves x(x − 2) = 0 by dividing by x and reports only x = 2. Repair the reasoning and explain why division was unsafe.Written reasoning: review required

Solve rational equations without restoring excluded inputs — Algebra foundations: expressions, equations and roots

• Solve (x² − 4)/(x − 2) = 4.Automatically checked response; reasoning still matters
• Solve 1/x + 1/(x + 1) = 3/2.Automatically checked response; reasoning still matters
• Solve (x + 1)/(x − 3) = 4/(x − 3) + 1. Explain why saying every real number is inaccurate.Written reasoning: review required

Check roots introduced by squaring — Algebra foundations: expressions, equations and roots

• Solve √(x + 6) = x.Automatically checked response; reasoning still matters
• A learner squares √(x + 1) = x − 1, finds x = 0 and 3, and keeps both. Repair the solution and explain the logical issue.Written reasoning: review required

Solve and interpret inequalities — Algebra foundations: expressions, equations and roots

• Which set solves 7 − 3x > 16?Automatically checked response; reasoning still matters
• Solve −2 ≤ 3x + 1 < 10.Automatically checked response; reasoning still matters
• Explain why dividing an inequality by a negative number reverses its direction. Use −2x < 6 and one accepted and one rejected value to illustrate.Written reasoning: review required

Rearrange formulas with conditions — Algebra foundations: expressions, equations and roots

• Solve y = mx + b for x, including m = 0. Explain why x = (y − b)/m is not the whole answer.Written reasoning: review required

Solve inequalities with an unknown coefficient — Algebra reasoning, constraints and solution evidence

• Solve ax+6>0 for every real parameter a, explicitly including a=0.Written reasoning: review required
• Solve (a−2)x≥a−2 for every real a. Compare the result with canceling a−2 without a case split.Written reasoning: review required
• Classify ax≤b for arbitrary real a,b. Give all cases when a=0, then illustrate one empty and one all-real set.Written reasoning: review required

Classify and solve quadratic equations — Algebra foundations: expressions, equations and roots

• Solve 2x² − 5x − 3 = 0 over the real numbers.Automatically checked response; reasoning still matters
• For ax² + 4x + 4 = 0, explain the cases a = 1, a = 2 and a = 0. Give the roots in each case and state why the formula cannot be used unchanged when a = 0.Written reasoning: review required

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• For the new event n(p)=150−5p on [0,30], with fixed cost 100 dollars and 4 dollars per ticket, construct revenue, cost and profit functions with units. Compare the revenue-maximizing and profit-maximizing prices and their profits.Written reasoning: review required

Choose a quadratic method and derive the formula — Algebra reasoning, constraints and solution evidence

• Derive the quadratic formula for real ax²+bx+c=0 with a≠0. State why using 2a in the denominator is valid even for negative a.Written reasoning: review required
• Solve 2x²+12x+7=0 by completing the square, without quoting the formula. Verify your exact pair structurally.Written reasoning: review required

Factor structure and use zero products — Algebra foundations: expressions, equations and roots

• Solve x² − x − 12 = 0.Automatically checked response; reasoning still matters

Classify and solve quadratic equations — Algebra foundations: expressions, equations and roots

• Solve 2x² − 5x − 3 = 0 over the real numbers.Automatically checked response; reasoning still matters
• Give the distinct real roots of 9x² + 12x + 4 = 0.Automatically checked response; reasoning still matters
• Which gives both roots of x² − 4x + 1 = 0 exactly?Automatically checked response; reasoning still matters
• Over the complex numbers, what are the roots of x² + 2x + 5 = 0?Automatically checked response; reasoning still matters

Why every quadratic has two complex roots — Real numbers, complex coordinates, vectors and matrices

• For real a,b,c with a≠0, prove ax²+bx+c has exactly two complex roots counting multiplicity. Separate positive, zero and negative discriminants.Written reasoning: review required

Choose a quadratic method and derive the formula — Algebra reasoning, constraints and solution evidence

• Solve 2x²+12x+7=0 by completing the square, without quoting the formula. Verify your exact pair structurally.Written reasoning: review required
• Choose and justify efficient methods for x²=81, (x−4)(x+2)=0 and x²−6x+11=0, then solve all three over the complex numbers.Written reasoning: review required
• A learner solves (2x+1)²=16 by writing2x+1=4 only. Repair it. Explain what changes for right sides0 and −16 over real and complex numbers.Written reasoning: review required

Solve systems without changing their solutions — Linear Algebra I: systems, spaces and linear maps

• Prove replacing equation E2 by E2+kE1 preserves solutions when E1 is kept.Written reasoning: review required
• Which operation always preserves a system's solution set?Automatically checked response; reasoning still matters

Find intersections and classify systems — Algebra II and precalculus: functions and structure

• Solve 2x+y=11 and x−y=1. Enter the x-coordinate.Automatically checked response; reasoning still matters
• Solve 3x+2y=14 and x−y=3. Enter the y-coordinate.Automatically checked response; reasoning still matters
• Classify 2x+4y=6 and x+2y=5.Automatically checked response; reasoning still matters
• A fictional workshop uses 2 hours and 3 material units for each A, and 1 hour and 4 units for each B. Exactly 14 hours and 32 material units are used. Model and solve for whole nonnegative counts. Explain what the graph intersections mean.Written reasoning: review required

Solve systems without changing their solutions — Linear Algebra I: systems, spaces and linear maps

• Solve x+y=7 and 2x−y=5. Enter x.Automatically checked response; reasoning still matters

Read intersections as simultaneous solutions — Algebra reasoning, constraints and solution evidence

• In supplied Graph G1, A is y=2x+1 and B is y=−x+4. What is the x-coordinate of their intersection?Automatically checked response; reasoning still matters
• Read the full solution point from Graph G1, then verify it in both equations. Explain why reporting only x=1 is incomplete when asked for an ordered-pair solution.Written reasoning: review required
• Construct three points on each line y=3x−2 and y=3x+1 and use them to draw the lines on shared axes. Then compare the system y=3x−2, 2y=6x−4. How many solutions does each system have?Written reasoning: review required

Algebra reasoning: mixed independent evidence — Algebra reasoning, constraints and solution evidence

• Construct shared scaled graphs of y=(7/10)x+11/10 and y=−(2/5)x+37/10 using labeled points. Estimate their crossing, then give exact coordinates and a table bracket for its x-coordinate between 2.36 and 2.37. Explain the relation between the crossing and output equality.Written reasoning: review required

Find intersections and classify systems — Algebra II and precalculus: functions and structure

• For y=x² and y=x+2, enter every real x-coordinate of an intersection.Automatically checked response; reasoning still matters
• Solve the system x²+y²=25 and y=x+1 completely. Show the ordered pairs and verify them.Written reasoning: review required

Read intersections as simultaneous solutions — Algebra reasoning, constraints and solution evidence

• In supplied Graph G2, C is y=x² and D is y=4. How many intersection points are there?Automatically checked response; reasoning still matters
• Use Graph G2 to read both intersection points. Verify them and explain why drawing only the right side of the parabola would lose a solution.Written reasoning: review required
• Sketch x²+y²=10 and y=x with equal scales. Estimate the two crossings and derive their exact coordinates. Is the vertical-line test a reason to reject the circle as a graph of an equation?Written reasoning: review required

Solve systems without changing their solutions — Linear Algebra I: systems, spaces and linear maps

• Solve x+y+z=6, y+z=4, z=1, and check all original equations.Written reasoning: review required
• When subtracting one equation from another, what happens to the right-hand side?Automatically checked response; reasoning still matters

Multiply matrices in the order the maps act — Linear Algebra I: systems, spaces and linear maps

• For A=[[2,−1],[3,0]] and x=(4,5), enter the second coordinate of Ax.Automatically checked response; reasoning still matters

Encode and verify a matrix solution — Algebra reasoning, constraints and solution evidence

• Write the row for 2x−y+3z=4 using unknown order (z,x,y). What is the second coefficient?Automatically checked response; reasoning still matters
• Enter A=[[1,0,1],[1,1,0],[0,1,1]] and b=(3,1,2) into the inverse tool. Record A inverse and the solution, and independently check one full row of A times A inverse and all three original equations.Written reasoning: review required

Algebra reasoning: mixed independent evidence — Algebra reasoning, constraints and solution evidence

• Encode 2x+y=6, y+z=5, 2z=6 as Ax=b in order (x,y,z). Use the available inverse calculator or independently documented exact elimination to obtain A inverse and the solution. Verify the inverse by a full matrix product and the solution in all original equations. Then explain what changes if the last coefficient row becomes zero while its right side remains 6.Written reasoning: review required

Undo a square linear system when possible — Linear Algebra I: systems, spaces and linear maps

• For A=[[2,1],[0,3]], enter the (1,2) entry of A⁻¹.Automatically checked response; reasoning still matters
• Find and verify the inverse of [[1,1],[1,2]].Written reasoning: review required
• A=[[1,2],[0,1]]. Solve Ax=(9,3). Enter x₁.Automatically checked response; reasoning still matters
• Which matrix is singular?Automatically checked response; reasoning still matters

Which number laws survive for matrices? — Real numbers, complex coordinates, vectors and matrices

• Explain A+0=A and AI=IA=A for a 2×2 matrix. Prove a 2×2 matrix [[a,b],[c,d]] is invertible exactly when ad−bc≠0. Include a nonzero singular example and show why cancellation can fail.Written reasoning: review required

Encode and verify a matrix solution — Algebra reasoning, constraints and solution evidence

• Enter A=[[1,0,1],[1,1,0],[0,1,1]] and b=(3,1,2) into the inverse tool. Record A inverse and the solution, and independently check one full row of A times A inverse and all three original equations.Written reasoning: review required
• Compare x+y+z=2, 2x+2y+2z=4, x−y=0 with the system obtained by replacing the second right side with 5. Explain the inverse status and describe all solutions or prove none exist.Written reasoning: review required
• For A=[[1,1,0],[0,1,1],[1,0,1]], a learner proposes B=(1/2)[[1,1,−1],[−1,1,1],[1,−1,1]]. Test the first row of AB. Explain why a plausible-looking array or a solution for one right side does not prove that B is A inverse.Written reasoning: review required

Find intersections and classify systems — Algebra II and precalculus: functions and structure

• Solve the system x²+y²=25 and y=x+1 completely. Show the ordered pairs and verify them.Written reasoning: review required
• A fictional workshop uses 2 hours and 3 material units for each A, and 1 hour and 4 units for each B. Exactly 14 hours and 32 material units are used. Model and solve for whole nonnegative counts. Explain what the graph intersections mean.Written reasoning: review required

Read domains and ranges across representations — Algebra II and precalculus: functions and structure

• Which listed relation fails to define a function of its first coordinate?Automatically checked response; reasoning still matters

An equation describes a set of points — Algebra reasoning, constraints and solution evidence

• Prove that the solution set of 3x+2y=8 is exactly the line y=4−(3/2)x. State both implications, not just one substitution.Written reasoning: review required
• Sketch x²+y²=13 using equal coordinate scales. Label (2,3), (2,−3), (−2,3), and (−2,−3), the center and the axis intercepts. Explain why the upper semicircle alone is insufficient.Written reasoning: review required
• A learner says y=(x²−9)/(x−3) has the entire line y=x+3 as its graph. Give the exact correction and justify both inclusion directions.Written reasoning: review required
• Describe and draw the entire solution set of (x−2)(y+1)=0. Explain why testing five points cannot prove your description.Written reasoning: review required

Find intersections and classify systems — Algebra II and precalculus: functions and structure

• For y=x² and y=x+2, enter every real x-coordinate of an intersection.Automatically checked response; reasoning still matters

Solve target times and justify rounding — Functions in graphs, measurements and growth decisions

• For Q(t)=2·10^(0.3t), solve Q(t)=9 and find the first integer time at or above 9. Supply neighboring checks.Written reasoning: review required
• A concentration is C(t)=100e^(−0.2t) for t≥0 hours. Solve C(t)≤30 and find the first whole-hour report satisfying it.Written reasoning: review required

Turn a sign bracket into a certified root interval — Numerical analysis: algorithms, errors and evidence

• Prove bisection converges under continuity and opposite endpoint signs.Written reasoning: review required
• Why does 1/x on [−1,1] have no bisection root guarantee?Written reasoning: review required

Approximate intersections without inventing solutions — Algebra reasoning, constraints and solution evidence

• For f(x)=x³ and g(x)=x+2, compute h=f−g at 1.52 and 1.53. Give a root bracket and justify why there is only one root in that bracket. Do not claim a rounded value is exact.Written reasoning: review required
• Solve 1/(x−2)=x. Give exact roots and approximate coordinates of the intersections with y=x. Explain why a sign change across x=2 is not a root certificate.Written reasoning: review required
• Find all intersections of y=|x−1| and y=x/2+2. Draw the two branches of the V and verify each answer in the original equation.Written reasoning: review required
• For 2^x=3 use the supplied calculator values 2^1.58≈2.9896985 and 2^1.59≈3.0104935. State a bracket, justify uniqueness, and distinguish a solution to 0.01 accuracy from an exact symbolic answer.Written reasoning: review required
• Determine every intersection of y=ln x and y=1−x. State the domain and prove completeness without relying on a finite plot.Written reasoning: review required

Read a linear objective on a polyhedron — Optimization: models, certificates and algorithms

• Maximize 3x+2y with x,y≥0, x+y≤4 and x≤2. What is the optimum value?Automatically checked response; reasoning still matters
• Explain why replacing an equality by a one-sided inequality can change a resource-allocation model.Written reasoning: review required

Build a feasible region from constraints — Algebra reasoning, constraints and solution evidence

• Draw the region x≥0, y≥0, x+y≤5, y>x−1. Label each boundary solid or dashed, give an interior point and decide whether (3,2) belongs.Written reasoning: review required
• Compare x+y≤2 together with x+y>2, and x≥0 together with y≥1. Identify empty or unbounded feasible sets and prove each statement.Written reasoning: review required
• A school makes kits x and posters y, both whole numbers. A kit uses 2 sheets and 3 minutes; a poster uses 1 sheet and 2 minutes. There are 16 sheets and 27 minutes, and at least 3 kits are required. Form the feasible system and decide whether (5,6) and (6,4) are allowed.Written reasoning: review required
• A graphing window shows a shaded region stopping at x=6. The only constraints are x≥1 and y≥x. Explain why calling x=6 a maximum is invalid, and give a feasible point beyond the window.Written reasoning: review required