Solving a Quadratic Equation Using the Quadratic Formula
(KS4, Year 10)
How to Solve Quadratic Equations Using the Quadratic Formula
Solving a quadratic equation using the quadratic formula is easy.Question
Solve the quadratic equation shown below using the quadratic formula.Step-by-Step:
1
2
3
$$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(2)}}{2(2)}$$
$$\:\:\:\: = \frac{5 \pm \sqrt{(-5 \times -5) - 4 \times 2 \times 2}}{2 \times 2}$$
$$\:\:\:\: = \frac{5 \pm \sqrt{25 - 16}}{4}$$
$$\:\:\:\: = \frac{5 \pm \sqrt{9}}{4}$$
$$\:\:\:\: = \frac{5 \pm 3}{4}$$
4
$$x = \frac{5 + 3}{4}$$
$$\:\:\:\: = \frac{8}{4}$$
$$\:\:\:\: = 8 \div 4$$
$$\mathbf{x = 2}$$
5
$$x = \frac{5 - 3}{4}$$
$$\:\:\:\: = \frac{2}{4}$$
$$\mathbf{x = \frac{1}{2}}$$
Answer:
We have solved the quadratic equation: x = ^{1}⁄_{2}, x = 2.2 Roots
Quadratic equations have 2 roots, and the quadratic equation finds both of them. Look closely at the formula, and you'll see a ± sign:$$x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$$
$$x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$$
The Discriminant
The term in the formula that appears in a square root is called the discriminant:- b^{2} − 4ac > 0: there are 2 real, distinct roots.
- b^{2} − 4ac = 0: there is one repeated root.
- b^{2} − 4ac < 0: there are 2 complex roots.
Beware
Be Careful with Signs
a, b and c may be negative. Make sure you remember this when inserting them into the equation - write them inside brackets if need be.Worksheet
This test is printable and sendable