Worked-example math problems
92 problems with complete step-by-step solutions. Each problem is solved by the same engine that powers the interactive solvers — every step is generated, not hand-written.
Algebra
Solve x² − 3x + 2 = 0 Solve a quadratic equation by computing the discriminant and applying the quadratic formula. Solve 2x² + 5x − 3 = 0 Solve a quadratic equation with a leading coefficient other than 1. Solve x² + 4 = 0 Solve a quadratic equation that has complex conjugate roots. Solve 3x + 7 = 22 Isolate the unknown in a linear equation step by step. Solve 2x − 9 = 5 Solve a linear equation with subtraction on the left-hand side. Find the slope of the line through (2, 3) and (5, 9) Compute the slope, y-intercept and equation of the line through two points. Distance from (1, 2) to (4, 6) Use the distance formula to find the length between two points. Find the 20th term of 3, 7, 11, 15, … Use the nth-term formula for an arithmetic sequence with first term 3 and common difference 4. Sum of the first 8 terms of 2, 6, 18, 54, … Use the partial-sum formula for a geometric sequence with ratio 3. Evaluate log₂(64) Use the change-of-base formula to compute a logarithm with base 2. Factor x² − 5x + 6 Factor a quadratic trinomial into two binomials. Solve 2x + 3 < 7 Solve a linear inequality and write the solution as an interval. Solve x² − 5x + 6 ≤ 0 Solve a quadratic inequality using its roots and a sign chart. Simplify (x + 1)(x − 2) + x² Expand the product and collect like terms. Partial fractions of 1/(x² − 5x + 6) Decompose a rational function into a sum of simpler fractions.
Trigonometry
Evaluate sin(30°) Convert the angle to radians and evaluate the sine. Evaluate cos(60°) Convert the angle to radians and evaluate the cosine. Evaluate tan(45°) Compute the tangent of a special angle. Solve the right triangle with legs 5 and 12 Find the hypotenuse with the Pythagorean theorem and both acute angles with arctangent. Find the hypotenuse with legs 6 and 8 Apply a² + b² = c² to find the missing hypotenuse. Find the missing leg when c = 13 and one leg is 5 Rearrange a² + b² = c² to solve for the unknown leg. Law of cosines: a = 7, b = 10, C = 60° Use the law of cosines to find the third side and the remaining two angles. Law of sines: A = 30°, B = 60°, a = 10 Find the third angle and the two remaining sides with the law of sines. Convert 180° to radians Convert degrees to radians and find the reference angle and quadrant. Convert π/3 rad to degrees Convert a radian measure to degrees and report the reference angle. Solve sin(x) = 1/2 Find every solution of a trigonometric equation, with the general solution. Solve tan(x) = 1 Solve a tangent equation and express the general solution. Convert (3, 4) to polar coordinates Find the radius and angle of a point given in Cartesian coordinates.
Calculus
Find the derivative of x³ Apply the power rule to differentiate a single term. Find the derivative of 3x² − 5x + 2 Differentiate a quadratic polynomial term by term. Find ∫ (2x + 3) dx Apply the reverse power rule to compute an indefinite integral. Find ∫₀³ x² dx Compute a definite integral with the fundamental theorem of calculus. Find the derivative of eˣ at x = 0 Use a numeric derivative to confirm that the derivative of the exponential at 0 is 1. Find lim x→0 sin(x)/x The classic limit that underpins the derivative of the sine function. Find lim x→1 (x² − 1)/(x − 1) A removable discontinuity — factor and simplify, or sample both sides. Find the tangent line to y = x² at x = 3 Evaluate f(a) and f'(a) and assemble the tangent line equation. Find ∫₀^π sin(x) dx Approximate the area under one full positive arch of the sine curve. Find the derivative of sin(x)·x² at x = 1 Use a numeric central-difference derivative on a product of trig and polynomial. Differentiate x³ − 3x Apply the power rule term by term to find the derivative. Integrate 2x + 1 Find the indefinite integral with the reverse power rule. Tangent line to y = x² at x = 2 Find the equation of the tangent line at a given point.
Precalculus
Expand (x + 1)⁵ Apply the binomial theorem with whole-number exponent 5. Expand (2x − 1)³ Apply the binomial theorem with a non-unit leading coefficient. Compute the permutations 7P3 Count the number of ordered arrangements of 3 items chosen from 7. Compute the combinations 10C4 Count the number of unordered selections of 4 items from 10. Multiply (3 + 2i)·(1 − i) Use the FOIL method and i² = −1 to multiply two complex numbers. Divide (1 + i) / (1 − i) Multiply numerator and denominator by the conjugate to divide two complex numbers. Divide (x³ − 1) by (x − 1) Polynomial long division reveals x − 1 as a factor of x³ − 1. Solve 2ˣ = 32 Use logarithms to isolate the unknown exponent. Solve 3ˣ = 27 An exponential equation that resolves to a whole-number exponent. Expand (3x + 2)⁴ Apply the binomial theorem with both coefficients larger than 1. Find the inverse of f(x) = 2x + 3 Swap x and y and solve to find the inverse function. Compose f(x) = x² and g(x) = x + 1 Build the composition (f ∘ g)(x) by substitution. Equation of a circle with center (0, 0) and radius 5 Write the standard and expanded equations of a circle. Properties of the ellipse x²/25 + y²/9 = 1 Find the center, foci and eccentricity of an ellipse.
Statistics
Mean, median and mode of {2, 4, 4, 6, 8} Compute the central-tendency statistics of a small data set with a repeated value. Standard deviation of {1, 2, 3, 4, 5} Find the population and sample standard deviations of a uniform data set. z-score for x = 85, μ = 70, σ = 10 Standardize a single value to standard-deviation units. P(X ≤ 115) for IQ with μ = 100, σ = 15 Find the cumulative probability for a normal distribution modelling IQ scores. 95% confidence interval — x̄ = 50, s = 8, n = 40 Build a 95% confidence interval around a sample mean. Linear regression for (1, 2), (2, 4), (3, 5), (4, 4), (5, 6) Fit the least-squares line and report the correlation coefficient r. z-score for x = 72, μ = 80, σ = 4 A negative z-score puts the value two standard deviations below the mean. Mean of {10, 20, 30, 40, 50} Compute the mean of an evenly spaced data set. 99% confidence interval — x̄ = 120, s = 15, n = 100 A larger sample and a higher confidence level — what changes? P(Z ≤ 1.96) for the standard normal The classic critical value that gives roughly 97.5% of the standard normal mass. Binomial probability: P(X = 5) for n = 10, p = 0.5 Compute a binomial point probability and the cumulative tails. Poisson probability: P(X = 2) with λ = 3 Compute a Poisson point probability and its cumulative tails. Bayes' theorem: probability of disease given a positive test Update a low base rate with a positive test using Bayes' theorem. Z-test for a proportion: p̂ = 0.58, n = 200 Test a sample proportion against p₀ = 0.5 with a two-tailed z-test.
Linear Algebra
Determinant of [[3, 1], [4, 2]] Apply the 2×2 determinant formula det = ad − bc. Determinant of the 3×3 matrix [[1, 2, 3], [4, 5, 6], [7, 8, 10]] Use cofactor expansion along the first row. Multiply [[1, 2], [3, 4]] · [[5, 6], [7, 8]] Combine rows of A with columns of B to compute the product. Inverse of [[4, 7], [2, 6]] Check that the determinant is non-zero, then apply the 2×2 inverse formula. Inverse of [[2, 1, 1], [1, 3, 2], [1, 0, 0]] Use Gauss-Jordan elimination on the augmented [A | I] matrix. Add [[1, 2], [3, 4]] + [[5, 6], [7, 8]] Add two matrices of the same shape entry by entry. Dot product (1, 2, 3) · (4, 5, 6) Multiply matching components and sum to find the dot product. Cross product (1, 0, 0) × (0, 1, 0) Two standard basis vectors give the third one via the right-hand rule. Magnitude of the vector (3, 4) The Pythagorean theorem gives the length of a 2D vector. Angle between (1, 0) and (1, 1) Use the dot product to find the angle between two 2D vectors. Transpose of [[1, 2, 3], [4, 5, 6]] Swap the rows and columns of a 2×3 matrix. Eigenvalues and eigenvectors of [[4, 1], [2, 3]] Find the eigenvalues from the characteristic polynomial and an eigenvector for each. Solve 2x + y = 5, x + 3y = 10 Solve a 2×2 linear system with Gauss-Jordan elimination. Project (3, 4) onto (1, 0) Find the vector projection and the perpendicular component.
Finite Math
Compound interest on $1000 at 5% for 10 years Find the future value with monthly compounding. Simple interest on $1000 at 5% for 3 years Find the interest and final amount with the I = Prt formula. Monthly payment on a $200,000 mortgage at 5% for 30 years Find the level payment and total interest on an amortized loan. Steady state of a two-state Markov chain Iterate the distribution and solve for the steady-state vector. Maximise 5x + 4y subject to linear constraints Solve a two-variable linear program by the graphical method. Truth table: (p → q) ↔ (¬q → ¬p) Build a truth table to verify that a conditional and its contrapositive are equivalent.