What is the quadratic formula? — x = (−b ± √(b²−4ac)) / 2a — used to solve ax² + bx + c = 0. The discriminant (b²−4ac): > 0 = two real roots, = 0 = one real root, < 0 = no real roots.

What is the discriminant and what does it tell you? — Δ = b² − 4ac in a quadratic equation. Δ > 0 → two distinct real roots. Δ = 0 → one repeated real root. Δ < 0 → no real roots (two complex roots).

What is the slope-intercept form of a line? — y = mx + b where m = slope (rise/run) and b = y-intercept (where the line crosses the y-axis).

What is the point-slope form of a line? — y − y₁ = m(x − x₁) where m is the slope and (x₁, y₁) is a known point on the line. Useful when you know the slope and one point.

What is the standard form of a linear equation? — Ax + By = C where A, B, and C are integers and A ≥ 0. Useful for finding intercepts: set x=0 for y-intercept, y=0 for x-intercept.

How do you find the slope between two points? — m = (y₂ − y₁) / (x₂ − x₁) — rise over run. Positive slope = upward, negative = downward, zero = horizontal, undefined = vertical line.

What are parallel and perpendicular slopes? — Parallel lines: same slope (m₁ = m₂). Perpendicular lines: slopes are negative reciprocals: m₁ × m₂ = −1 (e.g., slopes of 2 and −½).

What are the laws of exponents? — Product: aᵐ × aⁿ = aᵐ⁺ⁿ. Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Power of power: (aᵐ)ⁿ = aᵐⁿ. Zero exponent: a⁰ = 1. Negative exponent: a⁻ⁿ = 1/aⁿ.

What is a negative exponent? — a⁻ⁿ = 1/aⁿ. It means the reciprocal. Example: 2⁻³ = 1/2³ = 1/8. A negative exponent does NOT make the number negative — it flips it.

What is a fractional exponent? — aᵐ/ⁿ = ⁿ√(aᵐ) — the denominator is the root, the numerator is the power. Example: 8²/³ = (³√8)² = 2² = 4.

What is the difference of squares factoring formula? — a² − b² = (a + b)(a − b). Example: x² − 9 = (x+3)(x−3). Only works when both terms are perfect squares and separated by subtraction.

What are perfect square trinomials? — a² + 2ab + b² = (a+b)² and a² − 2ab + b² = (a−b)². Example: x² + 6x + 9 = (x+3)². Recognize them when the constant is (b/2)².

How do you complete the square? — For x² + bx + c = 0: 1. Move c to right side. 2. Add (b/2)² to both sides. 3. Factor left side as (x + b/2)². 4. Take square root of both sides. Example: x²+6x=7 → (x+3)²=16 → x=1 or x=−7.

What is the FOIL method? — A method to multiply two binomials: First, Outer, Inner, Last. Example: (x+2)(x+3) = x²+3x+2x+6 = x²+5x+6.

What is a function in mathematics? — A relation where each input (x) has exactly one output (y). Written as f(x). The vertical line test determines if a graph is a function — a vertical line hits the curve at most once.

What is function composition? — (f ∘ g)(x) = f(g(x)) — apply g first, then apply f to the result. Example: if f(x)=x² and g(x)=x+1, then f(g(x)) = (x+1)². Order matters: f∘g ≠ g∘f in general.

What is an inverse function? — A function f⁻¹(x) that reverses f(x). If f(a)=b, then f⁻¹(b)=a. To find: swap x and y, then solve for y. The graph is a reflection of f(x) across the line y=x.

What is a logarithm? — logₐ(b) = c means aᶜ = b. The logarithm answers: "what power must a be raised to, to get b?" Example: log₂(8) = 3 because 2³ = 8.

What are the log rules? — Product: log(xy) = log(x)+log(y). Quotient: log(x/y) = log(x)−log(y). Power: log(xⁿ) = n·log(x). Change of base: logₐ(x) = log(x)/log(a).

What is the natural logarithm (ln)? — ln(x) = logₑ(x) — logarithm with base e ≈ 2.718 (Euler's number). ln(eˣ) = x and e^(ln x) = x. ln(1) = 0 and ln(e) = 1.

How do you solve systems of equations by substitution? — 1. Solve one equation for one variable. 2. Substitute that expression into the other equation. 3. Solve for the remaining variable. 4. Substitute back to find the first variable.

How do you solve systems of equations by elimination? — 1. Multiply equations to make one variable's coefficients equal. 2. Add or subtract equations to eliminate that variable. 3. Solve for the remaining variable. 4. Substitute back.

What is the arithmetic sequence formula? — aₙ = a₁ + (n−1)d where aₙ = nth term, a₁ = first term, d = common difference. Sum of n terms: Sₙ = n/2 × (a₁ + aₙ). Example: 2, 5, 8, 11… d=3.

What is the geometric sequence formula? — aₙ = a₁ × rⁿ⁻¹ where r = common ratio. Sum of n terms: Sₙ = a₁(1−rⁿ)/(1−r) (r≠1). Infinite sum (

What is the binomial theorem? — (a+b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ for k=0 to n. Coefficients come from Pascal's Triangle. Example: (a+b)² = a² + 2ab + b². (a+b)³ = a³ + 3a²b + 3ab² + b³.

What are inequalities and how do you solve them? — Same rules as equations EXCEPT: when multiplying or dividing by a negative number, flip the inequality sign. Example: −2x < 6 → x > −3.

What is the distance formula between two points? — d = √((x₂−x₁)² + (y₂−y₁)²) — derived from the Pythagorean theorem. Gives the straight-line distance between (x₁,y₁) and (x₂,y₂).

What is the midpoint formula? — M = ((x₁+x₂)/2, (y₁+y₂)/2) — the average of the x-coordinates and the average of the y-coordinates.