Quadratics and Factoring
Solving quadratic equations by factoring, completing the square, and the quadratic formula — and what the solutions reveal about the parabola.
The Quadratic
A quadratic is a polynomial of degree 2:
ax² + bx + c = 0 (a ≠ 0)
It appears everywhere — projectile motion, area problems, optimisation, signal processing. Understanding quadratics is understanding how quantities that accelerate or curve behave.
The graph of y = ax² + bx + c is a parabola:
- Opens up if a > 0, down if a < 0
- Has a single turning point — the vertex
- Is symmetric about the vertical line through the vertex
Factoring
Factoring rewrites a quadratic as a product of two linear terms. If you can do it, it’s the fastest method.
Common factor first
Always check for a common factor before anything else:
6x² + 9x = 3x(2x + 3)
Factoring x² + bx + c (a = 1)
Find two numbers that multiply to c and add to b:
x² + 7x + 12
→ need two numbers: product = 12, sum = 7 → 3 and 4
→ (x + 3)(x + 4)
x² − 5x + 6
→ product = 6, sum = −5 → −2 and −3
→ (x − 2)(x − 3)
Factoring ax² + bx + c (a ≠ 1)
Multiply a × c, find two numbers that multiply to ac and add to b, split the middle term:
2x² + 7x + 3
a × c = 6, sum = 7 → 6 and 1
2x² + 6x + x + 3
= 2x(x + 3) + 1(x + 3)
= (2x + 1)(x + 3)
Special patterns — memorise these
Difference of squares: a² − b² = (a + b)(a − b)
Perfect square (sum): a² + 2ab + b² = (a + b)²
Perfect square (diff): a² − 2ab + b² = (a − b)²
Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²)
Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²)
x² − 16 = (x + 4)(x − 4)
x² + 6x + 9 = (x + 3)²
4x² − 12x + 9 = (2x − 3)²
Solving by Factoring
Set each factor to zero — if ab = 0, then a = 0 or b = 0:
x² − 5x + 6 = 0
(x − 2)(x − 3) = 0
x = 2 or x = 3
These are the roots — where the parabola crosses the x-axis.
Completing the Square
Any quadratic can be rewritten in vertex form: a(x − h)² + k.
x² + 6x + 5
= (x² + 6x + 9) − 9 + 5 (add and subtract (6/2)² = 9)
= (x + 3)² − 4
Vertex: (−3, −4). Minimum value: −4 at x = −3.
To solve by completing the square:
x² + 6x + 5 = 0
(x + 3)² = 4
x + 3 = ±2
x = −1 or x = −5
Completing the square is how the quadratic formula is derived.
The Quadratic Formula
For any quadratic ax² + bx + c = 0:
x = (−b ± √(b² − 4ac)) / 2a
Always works. Use it when factoring isn’t obvious.
2x² − 4x − 6 = 0
a = 2, b = −4, c = −6
x = (4 ± √(16 + 48)) / 4
= (4 ± √64) / 4
= (4 ± 8) / 4
→ x = 3 or x = −1
The Discriminant
The expression b² − 4ac under the square root is the discriminant (Δ).
| Δ | Roots | Parabola |
|---|---|---|
| Δ > 0 | two distinct real roots | crosses x-axis twice |
| Δ = 0 | one repeated real root | touches x-axis once (vertex on axis) |
| Δ < 0 | no real roots (two complex) | doesn’t cross x-axis |
x² + 2x + 5: Δ = 4 − 20 = −16 < 0 → no real roots
x² − 2x + 1: Δ = 4 − 4 = 0 → one root: x = 1
x² − 3x + 2: Δ = 9 − 8 = 1 > 0 → two roots
The discriminant tells you the nature of solutions without solving.
The Vertex
The vertex is the turning point — minimum if a > 0, maximum if a < 0.
x-coordinate of vertex:
x = −b / 2a
(Midpoint of the two roots, if they exist.)
y-coordinate: substitute back into the equation.
f(x) = x² − 4x + 1
vertex x: −(−4)/(2×1) = 2
vertex y: 4 − 8 + 1 = −3
vertex: (2, −3)
Vertex form: y = a(x − h)² + k, vertex at (h, k). Useful for reading off the minimum/maximum directly.
Vieta’s Formulas
For ax² + bx + c = 0 with roots r and s:
r + s = −b/a (sum of roots)
r × s = c/a (product of roots)
You can often find roots or verify solutions without solving:
x² − 7x + 12 = 0
sum of roots = 7, product = 12 → roots are 3 and 4
Quadratics in Disguise
Some equations become quadratic with a substitution:
x⁴ − 5x² + 4 = 0
let u = x²:
u² − 5u + 4 = 0
(u − 1)(u − 4) = 0
u = 1 or u = 4
x² = 1 or x² = 4
x = ±1 or x = ±2
Recognising this pattern — a “biquadratic” — avoids unnecessary complexity.