Quadratic Equations

Standard form

ax² + bx + c = 0

Method 1 — factorising

x² + 5x + 6 = 0
(x + 3)(x + 2) = 0

A product is zero only if a bracket is zero, so:

x + 3 = 0 → x = −3 or x + 2 = 0 → x = −2.

Method 2 — the quadratic formula

When it won't factorise, use:

        −b ± √(b² − 4ac)
x  =  ─────────────────────
              2a

Solve 2x² + 3x − 5 = 0 (a=2, b=3, c=−5):

x = [ −3 ± √(9 + 40) ] / 4
x = [ −3 ± √49 ] / 4
x = (−3 ± 7) / 4
x = 1   or   x = −2.5

Method 3 — completing the square

x² + 6x + 4 → (x + 3)² − 9 + 4 → (x + 3)² − 5.

Useful for finding the turning point of a parabola: vertex at (−3, −5).

The discriminant b² − 4ac

• > 0 → two real solutions
• = 0 → one repeated solution
• < 0 → no real solutions

Exam tip

After factorising, set each bracket = 0. There are usually two answers — don't stop at one. The formula is on most formula sheets, but learn it anyway.