Math guides

Worked examples for the algebra, precalculus, trigonometry, calculus, linear algebra and probability topics where students most often lose points. Each guide shows every step, the common mistake, and the exact line where it happens, with practice problems and answers at the end. Written for students, and for parents helping with homework.

• Completing the square, step by step

  Solve a quadratic or find vertex form by building a perfect square, including what to do when the x² term has a coefficient.

• Factoring trinomials when a ≠ 1 (the AC method)

  Factor ax² + bx + c by splitting the middle term and grouping, and spot the sign slip that makes the groups not match.

• Why (a + b)² isn't a² + b² — and other distribution mistakes

  Why the middle term matters, shown with numbers and a picture, plus the minus-sign and square-root versions of the same mistake.

• Solving equations with fractions: clear the denominators first

  Multiply every term by the least common denominator, then solve an equation with no fractions in it. Includes the term people forget to multiply.

• Flipping the inequality sign: when and why

  Flip only when you multiply or divide by a negative. Why that's the rule, and the two ways students flip when they shouldn't.

• Solving systems of equations: substitution vs elimination

  How to pick the method that keeps the arithmetic short, both worked in full, the no-solution case, and the dropped parenthesis that ruins most substitutions.

• Negative and zero exponents: what they mean and the mistakes to avoid

  Why x⁰ = 1 and x⁻ⁿ = 1/xⁿ, shown as a pattern, and the five ways they get misapplied, starting with 2⁻³ = −8.

• Quadratic formula sign mistakes: where they happen and how to catch them

  Six sign errors, each on the line it lives on, and two ten-second checks that tell you which of a, b, or c went wrong.

• Simplifying rational expressions: cancel factors, not terms

  Factor completely, cancel only what multiplies the whole top and bottom, keep the excluded values, and watch for (x − 3) over (3 − x).

• Absolute value equations: two cases, then check

  Isolate the bars, split into two cases, spot the no-solution case, and throw out the answer that fails when x is on both sides.

• Logarithm rules: what you can split apart, and what you can't

  The product, quotient, and power rules, why log(x + y) isn't log x + log y, and the domain check that removes a fake solution.

• Finding the equation of a line: slope, point-slope, and slope-intercept

  Find the slope from two points, write point-slope form, convert to y = mx + b, and get parallel and perpendicular slopes right.

• Absolute value inequalities: when it’s “and”, when it’s “or”

  Less-than means between, greater-than means outside. Isolate first, write the two cases correctly, and handle a negative right side.

• Special factoring: difference of squares, sum and difference of cubes

  Factor a² − b², a³ − b³ and a³ + b³, keep the cube sign pattern straight, take out the GCF first, and factor x⁴ − 16 all the way.

• Polynomial long division and synthetic division: the missing-term trap

  Divide polynomials by long division and synthetic division, with 0 placeholders for missing powers, the x + 2 → −2 rule, and the remainder theorem as a check.

• Fractional exponents: what x^(m/n) means and how to simplify

  Why x^(1/n) is an nth root and x^(m/n) is a root then a power, how negative fractional exponents work, and why 8^(2/3) isn't 16/3.

• Simplifying square roots and rationalizing the denominator

  Pull out the largest perfect square, combine only like radicals, rationalize with the conjugate, and why √(a + b) isn't √a + √b.

• Solving radical equations: isolate, square, then check

  Isolate the radical, square the whole other side, and check every answer, because squaring can create a solution that doesn't work.

• Solving exponential equations: same base, or take a log

  Rewrite with a common base when you can, take a log of both sides when you can't, and isolate the exponential before you do either.

• Composite and inverse functions: f(g(x)), f⁻¹(x), and the usual mix-ups

  Evaluate f(g(x)) from the inside out, find an inverse by swapping x and y, check it, and why f⁻¹(x) isn't 1/f(x).

• Complex numbers: i² = −1 and the mistakes it causes

  Roots of negatives, multiplying and dividing with i, powers of i, and why √(−4) · √(−9) is −6, not 6.

• Factoring by grouping: pair the terms, pull out the shared binomial

  Pair the terms, factor each pair, and pull out the shared binomial. Includes the sign flip in the second pair, the vanishing −1, and when to rearrange.

• Solving quadratics by factoring: zero on one side, then the zero-product property

  Get 0 on one side, factor, set each factor to 0. Why dividing by x loses a root, why you can't factor against a 15, and the sign flip when reading x from (x + 5).

• Rational equations: clear the denominators, then check for extraneous solutions

  List the values that zero a denominator, multiply every term by the LCD, solve, then throw out any answer on the list. Includes the equation with no solution.

• Domain and range: square roots, denominators, logs, and interval notation

  Square root: inside ≥ 0. Denominator: ≠ 0. Log: inside > 0. Interval notation done right, and the range of a square root or a parabola.

• Arithmetic and geometric sequences: the nth term, the sum, and the off-by-one in n

  The nth-term and sum formulas, why it's n − 1, the off-by-one when solving for n, a negative common ratio, and the sign in the geometric sum.

• Evaluating limits algebraically: plug in, factor, or use the conjugate

  Substitute first. When you get 0/0, factor and cancel or multiply by the conjugate; at infinity, divide by the highest power. Why 0/0 isn't the answer.

• Partial fraction decomposition: pick the right form, then find A and B

  Divide first if the top is too big, give a squared factor two terms and a quadratic Bx + C, then plug in roots. Includes the cover-up at the wrong root.

• Law of sines and law of cosines: which one to use, and the ambiguous case

  Which formula fits what you're given, the obtuse angle inverse sine misses, the second triangle in the SSA case, and radian mode.

• Proving trig identities: work on one side, and the moves that actually help

  Rewrite in sine and cosine, then use a Pythagorean identity, a common denominator, factoring, or a conjugate. Why working on both sides proves nothing.

• Solving trig equations: every solution, not just the calculator's

  Reference angle, then the two quadrants for the sign. Factor instead of dividing by sin x, stretch the interval for sin 2x, and find the answer arccos doesn't show.

• Area between two curves: top minus bottom, split where they cross

  Set the curves equal for the limits, integrate top minus bottom, split at every crossing, and integrate in y when the region is bounded by x = curves.

• Implicit differentiation: dy/dx on every y term, then solve for dy/dx

  Differentiate both sides, put dy/dx on every y term, use the product rule on xy, then collect and solve. Includes tangent lines and the second derivative.

• L'Hôpital's rule: check the form, then differentiate top and bottom separately

  Plug in first, use the rule only on 0/0 or ∞/∞, differentiate top and bottom on their own, and rewrite 0 · ∞ and 1^∞ first. Why the quotient rule gives the wrong limit.

• Related rates: write the equation, differentiate in time, then plug in

  Ladders, cones, ripples and planes: an equation true at every moment, the chain rule in t, and why plugging in the numbers too early triples the answer.

• Maxima and minima: critical points, endpoints, and the second-derivative test

  Critical points where f′ is zero or undefined, the endpoints of a closed interval, the first- and second-derivative tests, a fencing problem, and why a flat tangent isn't always a peak.

• The fundamental theorem of calculus: derivatives of integrals, and the chain rule on the limit

  Differentiate an integral by putting the limit in and multiplying by its derivative, flip the sign for a lower limit, evaluate with F(b) − F(a), and why ∫ 1/x² from −1 to 1 isn't −2.

• Volumes of revolution: disks, washers, and shells, and which one to use

  Disks, washers and shells: square each radius before subtracting, measure from the axis even when it's above the region, and keep the limits in the variable you integrate.

• Improper integrals: replace infinity with a limit, and check for blow-ups inside

  Replace ∞ with b, stop short of a blow-up with ε, split at an asymptote inside the interval, and use the p-test. Why 1/x → 0 isn't enough, and why ∫ x dx over the whole line isn't 0.

• Trig substitution: pick sin, tan, or sec from the square root, then draw the triangle

  x = a sin θ, a tan θ or a sec θ to clear the square root, the dx that comes with it, θ-limits for a definite integral, and reading the answer back off the triangle.

• Separable differential equations: separate, integrate, add C, then solve for y

  Move every y to one side, integrate, add C before solving for y, and match the sign to the initial condition. Why y = e^(sin x) + C fails, and why x + y won't separate.

• Riemann sums: left, right, midpoint, and trapezoid, and which way each one misses

  Left, right, midpoint and trapezoid sums: n strips but n + 1 points, x_i = a + iΔx, the half-weight ends, the limit that gives the integral, and why a left sum isn't always too small.

• u-substitution and integration by parts: the du you can't skip, and which part to call u

  Pick u so its derivative is already there, fix the constant in du, change the limits, and choose u and dv so the new integral is easier. Includes parts twice.

• Linear differential equations: standard form, the integrating factor, then integrate both sides

  Divide so y′ stands alone, take μ = e^(∫P dx) with P's sign, multiply both sides, and add C before dividing by μ. Why y = 6xe^(−3x) doesn't solve y′ + 3y = 6.

• Series convergence tests: which test to use, and what each one can't tell you

  The divergence test first, then geometric, p-series, comparison, integral, ratio or alternating. Why terms going to 0 prove nothing, and why bigger than a convergent series proves nothing either.

• Arc length and surface area: √(1 + (y′)²) under the integral, and 2πr for the surface

  Square y′, add 1, and look for a perfect square under the root. ds, not dx, for the width of a band, and the radius measured from the axis you turn around.

• Parametric and polar curves: dy/dx as a ratio of rates, and areas from ½r²

  dy/dx as (dy/dt)/(dx/dt), one more division by dx/dt for the second derivative, |speed| for arc length, and ½∫(r_out² − r_in²) dθ between polar curves.

• Gaussian elimination and matrix inverses: whole-row operations, and [A | I] to [I | A⁻¹]

  Row reduce the augmented matrix, clear below each pivot on the whole row, swap away a zero pivot, read a free variable or 0 = 1, and invert with [A | I] or the 2×2 formula. Why the 1/det can't be skipped.

• Eigenvalues and eigenvectors: det(A − λI) = 0, then solve (A − λI)v = 0

  The characteristic polynomial, the trace and determinant checks, each eigenvector as a null space, triangular matrices, a double root with one eigenvector, and powers by diagonalizing. Why a row of A − λI isn't the eigenvector.

• Bayes' rule: flip the conditional, and divide by the total probability

  Conditionals from a table, the law of total probability for the denominator, a positive test for a rare condition, the odds form, and a second test. Why a 95%-sensitive test can be right only 16% of the time.

• Expected value and variance of a discrete random variable: weight by the probabilities, then square before you average

  Weight each value by its probability, average the squares before subtracting the squared mean, scale the variance by a², and keep the binomial coefficient. Why linearity of expectation never needs independence.

• Continuous random variables: probability is area under the density, and the CDF does the integrating

  Normalize a density, get probabilities as areas or F(b) − F(a), find the mean by integrating, and use the uniform, exponential and normal. Why a density of 2 isn't a probability of 2.

• Double integrals: inner limits from the region, switching the order, and r dr dθ in polar

  Numbers on the outside, curves on the inside, redraw the region to switch the order, and never drop the r in r dr dθ. Includes the Gaussian integral and why squaring it needs a second variable.

• Product, quotient and chain rules: which rule when, and the inner derivative you can't drop

  Pick the rule from the last operation, put u′v first in the quotient rule, and multiply by the inner derivative, through rules nested inside rules. Why the derivative of a product isn't the product of the derivatives.

• Partial derivatives, gradients and Hessians: hold the other variables fixed, then read the whole Hessian

  Differentiate in one variable with the others frozen, get directional derivatives and the steepest rate from the gradient, classify critical points with f_xxf_yy − f_xy², and step downhill. Why the mixed partial can't be left out.

• Lagrange multipliers: ∇f = λ∇g, then let the constraint fix λ and compare every candidate

  Solve ∇f = λ∇g with the constraint, split into cases instead of canceling, compare every candidate on a closed curve, and read λ as a rate. Why a lone candidate on a line isn't automatically a maximum.

• Taylor series and linear approximation: divide by k!, expand in powers of x − a, and stay inside the radius

  Tangent-line estimates and the sign of their error, coefficients f⁽ᵏ⁾(a)/k! in powers of x − a, series from the geometric series, and remainder bounds. Why the bound takes the derivative's largest value.

• Gram–Schmidt and the QR decomposition: subtract the projections, normalize, then R = QᵀA

  Project with (a · b)/(a · a), subtract projections onto the q's and normalize, read coordinates off dot products, build A = QR, and solve least squares with Rx̂ = Qᵀb. Why QQᵀ isn't the identity for a tall Q.

• Second-order linear differential equations: the characteristic roots decide the form, and the constants come last

  The characteristic equation, distinct, repeated and complex roots, undetermined coefficients with the extra factor of t, and initial conditions applied to the full solution. Why a repeated root needs te^rt.

• Maximum likelihood estimation: take the log, set the derivative to zero, then check it's a maximum

  Write the log-likelihood, set its derivative to zero and check the second derivative, for exponential, geometric and normal data and readings with different noise. Why dividing by N − 1 doesn't give the MLE.

• Variance, covariance and correlation: center first, divide by N − 1, and square the constants

  Sample variance and covariance with N − 1, correlation from the standard deviations, Var(aX + bY) with its cross term, and a covariance matrix with aᵀΣa. Why Var(X − Y) isn't Var(X) − Var(Y).

• Least squares and the SVD: solve AᵀAx̂ = Aᵀb, check the residual, and read the rank off the singular values

  Fit a line with the normal equations, check Aᵀr = 0, compute an SVD by hand, find the shortest solution with the pseudoinverse, and keep σ₁u₁v₁ᵀ for the best rank-one approximation. Why singular values aren't eigenvalues.

• Entropy, cross-entropy and KL divergence: weight by p, keep the minus sign, and match the log to the unit

  Entropy in bits and nats and its perplexity, the cross-entropy loss of a batch with one-hot labels, KL(p ‖ q) as H(p, q) − H(p) checked against the definition, and the closed form for two Gaussians in both directions. Why KL(q ‖ p) isn't the same number as KL(p ‖ q), and why a negative entropy means a dropped minus sign.

• The multivariate Gaussian: invert Σ in the exponent, transform with AΣAᵀ, and condition with the Schur complement

  Evaluate the density through the squared Mahalanobis distance (x − μ)ᵀΣ⁻¹(x − μ), push it through a linear map with Aμ + b and AΣAᵀ, condition on one coordinate with the regression slope and the Schur complement, and sample with x = μ + Lε from a Cholesky factor. Why Var(x₁ − x₂) isn't Var(x₁) + Var(x₂) when the coordinates are correlated.

• Principal component analysis: center the data, take the eigenvectors of the covariance, and keep the top ones

  Subtract the mean, build the covariance matrix with N − 1, take its eigenvectors as the principal directions and its eigenvalues as the variances along them, compute scores and reconstructions, and count components from an SVD's squared singular values. Why skipping the centering tilts the top direction toward the mean.

• The central limit theorem and standard errors: divide σ by √N, standardize with the SE, and round the sample size up

  The standard error σ/√N of an average and σ√N for a sum, probabilities from the central limit theorem, a 95% confidence interval for a proportion, and the sample size for a target margin, rounded up. Why the spread of a sum isn't N times the spread of one value.

• Triple integrals: limits from the inside out, r dz dr dθ in cylindrical, and ρ² sin φ in spherical

  Inner limits that depend on the outer variables, r dz dr dθ under a slanted plane, the centroid of a cone, and ρ² sin φ for a hemisphere. Why dropping the r or the sin φ still gives a clean, wrong multiple of π.

• Vectors in 3-D: the dot product for angles, the cross product for normals, and |n| in every distance

  Angles and projections from the dot product, areas and normals from the cross product, the plane through three points, and the distance from a point to a plane with its nearest point. Why the middle entry of a cross product flips sign, and why every distance divides by |n|.

• Markov chains: multiply row vectors by P, solve πP = π, and get absorption from (I − Q)⁻¹

  The distribution after two steps as a row vector times P, the stationary distribution of a three-state chain with its return time, how fast the gap to it shrinks with the second eigenvalue, and expected years and graduation odds from the fundamental matrix. Why multiplying P by a column vector gives numbers that add up to more than 1.

• Line integrals and Green's theorem: ds = |r′(t)| dt, check for a potential first, and go counterclockwise

  The mass of a wire with ds = |r′(t)| dt, the work of a rotating field along a quarter circle, a potential function from the cross-partials test, and Green's theorem on a triangle checked side by side. Why the same field gives 4 along a segment and 2π along an arc.

• Quadratic forms and positive definite matrices: halve the cross term, test with eigenvalues or pivots, and read the axes off the eigenvectors

  The symmetric matrix of a form with its cross coefficient halved, three tests for positive definiteness (leading minors, completing the square as LDLᵀ and eigenvalues), the minors of a 3 × 3, and the axes and area of an ellipse. Why a positive diagonal proves nothing, and why the long axis belongs to the smallest eigenvalue.

• Conditional expectation and the law of total variance: average inside each slice, then add the spread between the slices

  The tower law for a die that decides how many coins to flip, the law of total variance on the same count, the mean and spread of two class sections with different means, and a stick broken twice. Why averaging the variances inside the groups leaves out the spread between them.

• Surface integrals and the divergence theorem: dS = |r_u × r_v| du dv, the normal sets the sign, and close the surface before using div F

  The area of a cone from √(1 + g_x² + g_y²), the outward flux through the side of a cylinder, the divergence theorem on the closed cylinder checked face by face, and Stokes' theorem on a paraboloid and its boundary circle. Why the divergence theorem gives the wrong flux for a surface that is not closed.

• Matrix exponentials and linear systems of ODEs: x(t) = e^Atx(0), diagonalize as Ve^ΛtV⁻¹, and read stability off the eigenvalues

  Solving x′ = Ax from the eigenvectors and the initial condition, the whole matrix e^At = Ve^ΛtV⁻¹, e^λt(I + Nt) for a repeated eigenvalue with one eigenvector, and a decaying spiral from complex eigenvalues. Why exponentiating each entry fails the first check at t = 0.

• Hypothesis tests and p-values: standardize under the null, choose the tails first, and never read p as the chance the null is true

  A one-sample z test on a class average, a test of an advertised 90% accuracy, a paired t test on before-and-after scores, and the share of significant results that are false alarms when most ideas do not work. Why a p-value is not the chance the null is true, and why ignoring the pairing hides a real effect.

• Laplace transforms: transform the equation with its initial values, solve for Y(s), then invert piece by piece from the table

  A transform from the table with the first shift theorem, an inverse with a repeated factor, a second-order initial-value problem, and an input that switches on at t = 1. Why dropping y(0) from the derivative rule solves a different problem, and why a delay must shift every t inside.

• Complex numbers in polar form: find the angle from the quadrant, multiply the lengths and add the angles

  The polar form of −√3 − i in the third quadrant, a product and a quotient checked in rectangular form, (√3 + i)⁵ by De Moivre, and the four fourth roots of −16. Why arctan(b/a) points the wrong way when a is negative, and why the root of a length is not the length divided by n.

• Counting and the binomial theorem: multiply the choices, divide out the orders that don't matter, and expand with C(n, k)

  The number of officer slates against plain committees from twelve people, the arrangements of MISSISSIPPI, the constant term of (x² − 2/x)⁶, and the chance of exactly 3 heads in 8 flips. Why a committee counted as 12 · 11 · 10 is six times too big, and why choosing one woman first overcounts “at least one”.

• Chi-square tests: compare counts with expected counts, divide each squared gap by E, and get the degrees of freedom right

  A spinner tested for fairness, offspring counts against a 9:3:3:1 ratio, a 2 × 3 table of grades by class section, and a 2 × 2 treatment table with the N(ad − bc)² shortcut. Why percentages halve the statistic for 200 spins, and why two different ratios can both pass.

• Fourier series: integrate against each sine and cosine, halve the constant term, and let symmetry do half the work

  The series of a step and its value at the jump, the cosine series of |sin x|, 1 − x² with period 2, and Parseval's identity giving 1 + 1/9 + 1/25 + ⋯ = π²/8. Why the constant term is a₀/2, and why cos nπ is (−1)ⁿ, never 0.

• How to convert a 4-point rubric score to a percentage (4 methods compared)

  For teachers: one paper scored 3, 3, 2 and 2 converted four published ways (straight percent, normalized percent as in Totara, Loma Linda University's anchored table and a Lake Washington school's chart), with a chart and table of every method from 1 to 4. Why a perfect paper must still come out at 100%, and why converting each score first can change the result.

• Matrix calculus conventions: numerator layout, gradients shaped like the variable, and the derivative read off the differential

  A 3 × 2 Jacobian from its entries, the gradient of (aᵀx)² from its differential, ∇_W(aᵀWb) = abᵀ, and the same derivative in both layouts. Why xᵀAx has gradient (A + Aᵀ)x, not 2Ax, unless A is symmetric, and why a matrix gradient must have the matrix's shape.

• Jacobians and the chain rule: outer Jacobian on the left, and backpropagate with the transpose

  A two-step composition multiplied out and checked, a backward pass through a tanh layer, ∇_WL = g xᵀ for the weights, and the count that makes backprop 500 times cheaper than multiplying from the input end. Why square matrices hide a wrong order or a missing transpose.

• The softmax Jacobian: diag(s) − ssᵀ, a backward pass without the matrix, and s − y for cross-entropy

  The softmax of (2, 1, 0) and its 3 × 3 Jacobian, the cross-entropy gradient s − y and one step that makes the target the most likely class, and Jᵀg = s ⊙ (g − (sᵀg)1) without forming J. Why the sigmoid's diagonal Jacobian doesn't carry over, and why e¹⁰⁰⁰ needs the largest score subtracted first.

• Regression gradients: (2/N)Xᵀ(Xw − y), Xᵀ(σ(Xw) − y)/N and Xᵀ(S − Y)/N, with the 1/N kept in view

  A three-point line fit and its gradient, the normal equations with and without ridge, a two-example logistic gradient and a softmax-regression gradient with its shapes. Why the mean squared error carries a 2/N, why ridge on a mean loss needs Nλ, and why no σ′ appears in the logistic gradient.

• Activation function derivatives: σ(1 − σ), 1 − tanh², the ReLU mask, and which ones you can compute from the output

  σ′ and tanh′ from the saved output, a ReLU and leaky ReLU backward pass, SiLU′ and GELU′ at 1, and the stable cross-entropy from a logit of 40. Why h(1 − h) needs h and not z, why the ReLU's mask isn't its output, and why SiLU's gate needs the product rule.

• Backprop for a one-hidden-layer network: δ₂ = ŷ − y, δ₁ = σ′(z₁) ⊙ W₂ᵀδ₂, and every gradient shaped like its weight

  A 2-2-2 sigmoid network from the forward pass to every gradient, then a batch of two with the 1/N applied once. Why the outer product is δhᵀ and not hδᵀ, why the hidden delta needs σ′, and why a batch's bias gradient sums the rows.

• Batched backprop for a dense layer: dW = XᵀdY, db = dYᵀ1, dX = dY Wᵀ, and the 1/N applied once

  A three-example batch from the forward pass to dW, db and dX, a mean squared loss checked against per-example outer products, and micro-batches of unequal size. Why the batch is the inner dimension of XᵀdY, why the 1/N goes in once, and why the input gradient needs Wᵀ.

• Layer norm backward: (g′ − mean(g′) − x̂ mean(g′ ⊙ x̂))/σ, and the two directions the loss can't see

  A four-feature token from the forward pass to the gradients of γ, β and x, then a batch of two tokens. Why μ and σ can't be held constant, why the gradient sums to 0 and is orthogonal to x̂, and why γ's gradient sums over tokens.

• Attention backward: AᵀG for V, a row-wise softmax for the scores, and the 1/√d on Q and K

  Two queries and two keys from the forward pass to the gradients of V, A, the scores, Q and K. Why ∇_VL needs Aᵀ, why each softmax row uses its own correction, and why the 1/√d stays in the gradients of Q and K.

• Batch norm backward: (dx̂ − mean(dx̂) − x̂ mean(dx̂ ⊙ x̂))/√(σ² + ε), one feature at a time down the batch

  A four-example, two-feature batch from the forward pass to the gradients of γ, β and X, then the same feature at inference. Why the batch statistics can't be held constant, why each column of dX sums to 0, and why the variance's derivative must divide by N.

• The normal distribution and z-scores: divide by σ, pick the right tail, and add the variances

  A battery's lifetime as three probabilities, a top-5% cutoff, the chance one model beats another and a conversion count with the continuity correction. Why z divides by σ, why a right tail is 1 − Φ, and why variances add even for a difference.

• Newton's method: x − f(x)/f′(x), digits that double, and the starts that run away

  √5 as exact fractions, a cubic with its error ratios, a minimum of x² + e⁻ˣ and one function from two starts. Why the step is subtracted, why a minimum needs f′/f″, and why a small f(x) is not proof of a root.

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