The quadratic formula
A quadratic equation ax² + bx + c = 0 with a ≠ 0 has solutions x = (−b ± √(b² − 4ac)) ÷ 2a.
The discriminant D = b² − 4ac decides the kind of roots: D > 0 gives two distinct real roots, D = 0 one repeated real root, and D < 0 two complex conjugate roots.
Example: x² − 5x + 6 = 0 has a = 1, b = −5, c = 6, so D = 25 − 24 = 1 and x = (5 ± 1) ÷ 2, giving x = 3 or x = 2.
| Quantity | Formula |
|---|---|
| Roots | x = (−b ± √(b² − 4ac)) ÷ 2a |
| Discriminant | D = b² − 4ac |
| Vertex | (h, k) with h = −b ÷ 2a, k = c − b² ÷ 4a |
| Vertex form | y = a(x − h)² + k |
| Factored form | y = a(x − x₁)(x − x₂) |
| Sum of roots | x₁ + x₂ = −b ÷ a |
| Product of roots | x₁ × x₂ = c ÷ a |
Sources: OpenStax: College Algebra 2e §2.5 Quadratic Equations
Special products and factoring
These identities hold for all real (and complex) a and b. Read left to right to expand; read right to left to factor.
| Name | Identity |
|---|---|
| Square of a sum | (a + b)² = a² + 2ab + b² |
| Square of a difference | (a − b)² = a² − 2ab + b² |
| Difference of squares | a² − b² = (a + b)(a − b) |
| Cube of a sum | (a + b)³ = a³ + 3a²b + 3ab² + b³ |
| Cube of a difference | (a − b)³ = a³ − 3a²b + 3ab² − b³ |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) |
| FOIL | (a + b)(c + d) = ac + ad + bc + bd |
| Trinomial square | (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca |
Sources: OpenStax: College Algebra 2e §1.4 Polynomials; OpenStax: College Algebra 2e §1.5 Factoring Polynomials
Lines, distance and properties of numbers
For points (x₁, y₁) and (x₂, y₂) in the plane:
- Slope: m = (y₂ − y₁) ÷ (x₂ − x₁)
- Slope-intercept form: y = mx + b
- Point-slope form: y − y₁ = m(x − x₁)
- Standard form: Ax + By = C
- Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²)
- Midpoint: ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
- Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1
- Distributive property: a(b + c) = ab + ac; commutative: a + b = b + a, ab = ba; associative: (a + b) + c = a + (b + c)
- Absolute value: |x| = x if x ≥ 0 and −x if x < 0; |ab| = |a||b|
Sources: OpenStax: College Algebra 2e §1.1 Real Numbers: Algebra Essentials