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Algebra · SS 1

Quadratic formula

Solve any quadratic equation, even one that won't factorise.

  • For ax2 + bx + c = 0:
  • x = −b ± √(b2 − 4ac)2a

What the letters mean

a
the number in front of x2 (not 0)
b
the number in front of x
c
the number on its own

Worked example

Solve 2x2 − 5x − 3 = 0.

  1. Step 1: Read off a, b and c

    a = 2, b = −5, c = −3

  2. Step 2: Work out b2 − 4ac

    (−5)2 − 4 × 2 × (−3) = 25 + 24 = 49

  3. Step 3: Put into the formula

    x = 5 ± √494 = 5 ± 74

  4. Step 4: Take + and − separately

    x = 124 = 3 or x = −24 = −12

Final answer: x = 3 or x = −12

Keep the signs: when b = −5, −b = +5.

See it

Play with this picture from the Quadratic Equations by Factorisation lesson.

−23

y = x2 − x − 6

x2 − x − 6 = (x + 2)(x − 3)

Roots: x = −2 or x = 3

Check: −2 + 3 = 1 (that's −b) and −2 × 3 = −6 (that's c)

b (the x number)−1
c (the number)−6