Worked methods
Math problem-solving guides
Identify the problem type, follow a transparent method, inspect common mistakes, and verify the result with a calculator or substitution check.
Algebra
How to Solve Linear Equations Step by Step
Solve one-variable linear equations by simplifying both sides, isolating the variable, and checking the result in the original equation.
3x + 7 = 22Result: x = 5Algebra
How to Solve Quadratic Equations
Choose factoring, completing the square, or the quadratic formula, then verify every root in the original quadratic equation.
x^2 - 5x + 6 = 0Result: x = 2 or x = 3Arithmetic
How to Simplify Fractions Correctly
Reduce a fraction by finding the greatest common divisor, dividing numerator and denominator by the same factor, and preserving the sign.
42/56 = 3/4Result: 3/4Applied math
How to Calculate Percentage Change
Calculate percentage increase or decrease from an old value to a new value, with the original value as the reference denominator.
(new - old) / |old| × 100%Result: 25% increaseCoordinate geometry
How to Find Slope Between Two Points
Find slope from two coordinates, interpret its sign, detect vertical lines, and verify the result using rise over run.
m = (y2 - y1) / (x2 - x1)Result: m = 3/2Statistics
How to Calculate Standard Deviation
Calculate population or sample standard deviation from deviations, squared deviations, variance, and the correct denominator.
s = sqrt(sum((xi - xbar)^2)/(n - 1))Result: s ≈ 2.14Linear algebra
How to Multiply Matrices
Multiply matrices by checking dimensions, taking row-by-column dot products, and verifying the shape and selected entries.
Cij = sum_k Aik BkjResult: AB = [[19,22],[43,50]]Calculus
How to Differentiate with the Power Rule
Differentiate polynomial terms using the power rule, constants, sums, and a substitution check against a difference quotient.
d/dx x^n = n x^(n-1)Result: f'(x) = 12x^3 - 4xCalculus
How to Integrate Basic Functions
Find basic antiderivatives term by term, include the constant of integration, and verify by differentiating the result.
∫ x^n dx = x^(n+1)/(n+1) + CResult: x^4/4 + 2x^2 + CAlgebra
How to Solve Systems of Linear Equations
Solve two-variable linear systems using elimination or substitution, classify the intersection, and check both original equations.
2x + y = 7; x - y = 2Result: x = 3, y = 1Algebra
How to Factor Quadratic Trinomials
Factor quadratic trinomials by identifying factor pairs, checking signs, and multiplying the factors back to the original polynomial.
x^2 + 7x + 12 = (x+3)(x+4)Result: (x+3)(x+4)Algebra
How to Solve Linear Inequalities
Solve one-variable linear inequalities, reverse the inequality when dividing by a negative, and represent the solution on a number line.
-2x + 3 > 11Result: x < -4Probability
How to Calculate Probability
Calculate probability from equally likely outcomes, complements, and conditional information while checking the sample space.
P(A) = favorable outcomes / total outcomesResult: P(A) = 1/6Geometry
How to Use the Pythagorean Theorem
Use the Pythagorean theorem for right triangles, identify the hypotenuse, solve for a missing side, and verify the geometry.
a^2 + b^2 = c^2Result: c = 5Scientific writing
How to Convert Scientific Notation
Convert between ordinary decimal form and scientific notation while preserving significant digits and the sign of the exponent.
0.00042 = 4.2 × 10^-4Result: 4.2 × 10^-4Financial mathematics
How to Calculate Compound Interest
Calculate compound growth from principal, nominal rate, compounding frequency, and time, then distinguish balance from interest earned.
A = P(1 + r/n)^(nt)Result: A ≈ 1104.94Coordinate geometry
How to Find the Equation of a Line
Find a line equation from a point and slope or two points, then convert between point-slope, slope-intercept, and standard form.
y - y1 = m(x - x1)Result: y = 2x + 1Linear algebra
How to Calculate a 2×2 Determinant
Calculate the determinant of a 2×2 matrix, interpret zero versus nonzero results, and verify signs and row operations.
det([[a,b],[c,d]]) = ad - bcResult: det = -2Statistics
How to Calculate Mean, Median, and Mode
Calculate and compare mean, median, and mode, then choose the measure that matches the distribution and question.
mean = sum(xi)/nResult: mean=5, median=4.5, mode=4Algebra
How to Solve Exponential Equations
Solve exponential equations by matching bases or taking logarithms, then check domain and substitute the result.
2^(x+1) = 16Result: x = 3How these guides are reviewed
Every guide states the method, includes worked examples, calls out failure cases, and gives a reproducible verification step. Calculators support checking; they do not replace assumptions or domain restrictions.
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