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- Harder quadratics: factorising, the formula and completing the square
Harder quadratics: factorising, the formula and completing the square
Not every quadratic factorises neatly. The quadratic formula and completing the square solve any of them.
Factorising ax2 + bx + c
Find two numbers that multiply to a × c and add to b, then split the middle term.
2x2 + 5x − 3: a × c = −6. Numbers: 6 and −1.
2x2 + 6x − x − 3 = 2x(x + 3) − (x + 3) = (2x − 1)(x + 3)
2x2 + 5x − 3: a × c = −6. Numbers: 6 and −1.
2x2 + 6x − x − 3 = 2x(x + 3) − (x + 3) = (2x − 1)(x + 3)
The quadratic formula
For ax2 + bx + c = 0:
x = (−b ± √b2 − 4ac)2a
Use it when the quadratic will not factorise, and round as the question asks.
x = (−b ± √b2 − 4ac)2a
Use it when the quadratic will not factorise, and round as the question asks.
Completing the square
x2 + bx + c = (x + b2)2 − (b2)2 + c
x2 + 6x + 2 = (x + 3)2 − 9 + 2 = (x + 3)2 − 7
The turning point is (−3, −7).
x2 + 6x + 2 = (x + 3)2 − 9 + 2 = (x + 3)2 − 7
The turning point is (−3, −7).
Solve x2 + 4x − 7 = 0, giving answers to 2 decimal places.
- a = 1, b = 4, c = −7
- x = (−4 ± √16 + 28)2 = (−4 ± √44)2
- x = 1.32 or x = −5.32
Answer: x = 1.32 or x = −5.32
Factorise ax2 + bx + c by splitting the middle term using a × c. Otherwise use x = (−b ± √(b2 − 4ac)) ÷ 2a. Completing the square gives the turning point.
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