Revision notes
Solve quadratic equations by factorising and setting each bracket to zero — and build an equation from its roots.
Yusuf's family has a rectangular garden that is 3 m longer than it is wide, with an area of 40 m². If the width is x, then x(x + 3) = 40. That is a quadratic equation — and factorising cracks it open.
A quadratic equation has an x2 term, like x2 + 3x − 40 = 0.
If two numbers multiply to give 0, at least one of them must be 0. So if (x − 5)(x + 8) = 0, then x − 5 = 0 or x + 8 = 0, giving x = 5 or x = −8.
That is why you must first make one side equal to 0 — the rule only works for 0.
🔑 Key idea: If A × B = 0, then A = 0 or B = 0.
Find two numbers that multiply to c and add to b. For x2 + 3x − 40: 8 × (−5) = −40 and 8 + (−5) = 3, so x2 + 3x − 40 = (x + 8)(x − 5).
In Yusuf's garden, x = 5 or x = −8. A width can't be negative, so the width is 5 m and the length is 8 m.
🔑 Key idea: Two numbers: product = c, sum = b.
For 2x2 + 5x − 3 = 0, multiply a × c: 2 × (−3) = −6. Two numbers with product −6 and sum 5 are 6 and −1.
Split the middle term and group: 2x2 + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (2x − 1)(x + 3). So x = 12 or x = −3.
Going backwards: roots 2 and −3 give (x − 2)(x + 3) = 0, which expands to x2 + x − 6 = 0. Notice the signs change.
🔑 Key idea: Roots p and q come from (x − p)(x − q) = 0.
Solve x2 + 7x + 12 = 0.
Step 1: Factorise
Two numbers that multiply to 12 and add to 7: 4 and 3
x2 + 7x + 12 = (x + 4)(x + 3)
Step 2: Set each bracket to zero
(x + 4) = 0 → x = −4
(x + 3) = 0 → x = −3
Answer: x = −4 or x = −3
Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.