Solving Quadratic Equations
Three methods for solving quadratics: factorising, the quadratic formula, and completing the square.
A quadratic equation can be written as ax² + bx + c = 0, where a is not zero. Its solutions are called roots.
Method 1: factorising
Use this when the equation factorises neatly.
Example. Solve x² + 5x + 6 = 0.
- Find two numbers that multiply to 6 and add to 5: 2 and 3.
- (x + 2)(x + 3) = 0.
- Either x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3.
- Check x = −2: 4 − 10 + 6 = 0. ✓ Check x = −3: 9 − 15 + 6 = 0. ✓
Method 2: the quadratic formula
For ax² + bx + c = 0:
x = (−b ± √(b² − 4ac)) ÷ (2a)
Example. Solve 2x² − 3x − 2 = 0, so a = 2, b = −3, c = −2.
- b² − 4ac = 9 − 4(2)(−2) = 9 + 16 = 25.
- √25 = 5.
- x = (3 ± 5) ÷ 4.
- x = 8/4 = 2 or x = −2/4 = −0.5.
- Check x = 2: 8 − 6 − 2 = 0. ✓ Check x = −0.5: 0.5 + 1.5 − 2 = 0. ✓
The discriminant
The value b² − 4ac tells you how many real roots there are:
- Greater than 0: two different real roots.
- Equal to 0: one repeated root.
- Less than 0: no real roots.
Method 3: completing the square
Rewrite x² + bx as (x + b/2)² − (b/2)². This method also gives the vertex of the parabola and is how the quadratic formula is derived.
Example. Solve x² + 6x + 2 = 0.
- (x + 3)² − 9 + 2 = 0.
- (x + 3)² = 7.
- x = −3 ± √7, approximately −0.35 or −5.65.
Common mistakes
- Forgetting to set the equation equal to zero before factorising.
- Losing the sign of b. Write b with its sign inside brackets when substituting.
- Forgetting the ± and giving only one solution.
- Dividing both sides by x, which can lose a solution.
Quick recap
Rearrange to equal zero. Try factorising. If that fails, use the formula. Check by substitution.