Solving a Quadratic Equation Using the Quadratic Formula
(KS4, Year 10)

The Lesson

The quadratic formula is a way of solving a quadratic equation. Consider a quadratic equation in standard form:

Solving the quadratic equation means finding the values of x, called roots, that make this equation true (i.e., makes the left hand side equal to 0.) The formula below will find the two roots of the equation:

There are two roots because the ± symbol means consider it as a + one time and as a another time.

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

Compare the quadratic equation in the question with the standard form.
2x2 −5x + 2 = 0 ⇔ ax2 + bx + c = 0
Find the values of a, b and c.
a = 2, b = −5, c = 2

2

Use the formula.
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

3

Substitute a, b and c into the formula. In our example, a = 2, b = −5 and c = 2.
$$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(2)}}{2(2)}$$ $$\:\:\:\: = \frac{5 \pm \sqrt{(-5 \times -5) - 4 \times 2 \times 2}}{2 \times 2}$$ $$\:\:\:\: = \frac{5 \pm \sqrt{25 - 16}}{4}$$ $$\:\:\:\: = \frac{5 \pm \sqrt{9}}{4}$$ $$\:\:\:\: = \frac{5 \pm 3}{4}$$

4

Find the root that comes from turning the ± into a +.
$$x = \frac{5 + 3}{4}$$ $$\:\:\:\: = \frac{8}{4}$$ $$\:\:\:\: = 8 \div 4$$ $$\mathbf{x = 2}$$
x = 2 is a root of the quadratic equation.

5

Find the root that comes from turning the ± into a −.
$$x = \frac{5 - 3}{4}$$ $$\:\:\:\: = \frac{2}{4}$$ $$\mathbf{x = \frac{1}{2}}$$
x = 12 is a root of the quadratic equation.

Answer:

We have solved the quadratic equation: x = 12, x = 2.

Lesson Slides

Sometimes quadratic equations have repeated roots: the same value of x solves the quadratic equation twice. The slider below shows another real example of how to solve a quadratic equation using the quadratic formula. Open the slider in a new tab

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:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
This means it is + one time, and − the other. This gives 2 roots:
$$x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$$ $$x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$$

The Discriminant

The term in the formula that appears in a square root is called the discriminant:
$$b^2 - 4ac$$
It discriminates between the 3 possible cases for the roots of a quadratic equation. We can visualize this by looking at a graph of a quadratic equation. The roots are the points where the curve crosses the horizontal x-axis.
  • b2 − 4ac > 0: there are 2 real, distinct roots.

  • b2 − 4ac = 0: there is one repeated root.

  • b2 − 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.
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See Also

What is a square number? What is a square root?