Quadratic Equations Cheat Sheet

Definition

The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Example: Example: 3x² + 2x - 1 = 0

Identifying Coefficients

In the standard form, 'a' is the coefficient of x², 'b' is the coefficient of x, and 'c' is the constant term.

Example: In 2x² - 5x + 3 = 0, a = 2, b = -5, c = 3

Leading Coefficient

The leading coefficient is the coefficient of the x² term (a). If a = 1, the quadratic is monic.

Example: In x² + 4x - 7 = 0, the leading coefficient is 1.

Factoring Method

Express the quadratic equation as a product of two binomials: (px + q)(rx + s) = 0. Then, set each factor equal to zero and solve for x.

Example: x² + 5x + 6 = (x + 2)(x + 3) = 0, so x = -2 or x = -3

Zero Product Property

If ab = 0, then a = 0 or b = 0 (or both). This is the basis for solving by factoring.

Example: If (x - 1)(x + 4) = 0, then x - 1 = 0 or x + 4 = 0

Difference of Squares

A special case: a² - b² = (a + b)(a - b)

Example: x² - 9 = (x + 3)(x - 3) = 0, so x = 3 or x = -3

The Formula

The solutions to ax² + bx + c = 0 are given by: x = (-b ± √(b² - 4ac)) / 2a

Example: For x² + 2x - 1 = 0, x = (-2 ± √(2² - 4(1)(-1))) / 2(1)

Discriminant

The discriminant (Δ = b² - 4ac) determines the nature of the roots:

Example: Δ > 0: two real roots; Δ = 0: one real root; Δ < 0: two complex roots

Real vs. Complex Roots

If the discriminant is positive or zero, the roots are real numbers. If the discriminant is negative, the roots are complex numbers.

Example: If b² - 4ac = -4, the roots are complex.