Solve any quadratic with the formula, round sensibly, and use b² − 4ac to count the solutions before you start.
What you will learn
- Solve ax² + bx + c = 0 with the formula
- Give answers to decimal places, significant figures or as surds
- Use the discriminant
When it will not factorise
Most quadratics do not factorise with whole numbers.
The formula always works:
x = (−b ± √(b² − 4ac)) ÷ 2a
- 2x² + 3x − 4 = 0: a = 2, b = 3, c = −4
- b² − 4ac = 9 − 4 × 2 × (−4) = 9 + 32 = 41
- x = (−3 ± √41) ÷ 4
- x = 0.85 or x = −2.35 (2 d.p.)
Write a, b and c down first, with their signs. If a is negative, as in −x² + 2x + 5 = 0, a = −1, and 2a = −2 on the bottom. If the equation is not = 0, rearrange it first: x(x + 3) = 7 becomes x² + 3x − 7 = 0.
A problem can lead to the formula. A stone's height is h = 15t − 5t². When is it 10 m high? 15t − 5t² = 10 gives 5t² − 15t + 10 = 0, so a = 5, b = −15, c = 10.
"Show that a solution is 2.41 to 2 decimal places" means work it out to more places first (say 2.4142) and round. Put negative numbers in brackets on the calculator: (−4), not −4 typed after a minus.
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- When it will not factorise
- Count the solutions first
- Exact answers
- Try it: See the formula worked out
7 practice questions, marked as you go; about 25 minutes in all.
In this unit: Quadratics
Expand and factorise quadratics, solve them by factorising, the formula and completing the square, and read roots and turning points from graphs.