How it works

When a quadratic equation ax² + bx + c = 0 does not factorise easily, the quadratic formula x = (-b ± √(b² - 4ac)) / 2a gives the exact solutions. The value b² - 4ac is called the discriminant, and its sign tells us how many real solutions the equation has: positive gives two, zero gives one repeated solution, and negative gives none.

The method

1. 1.Rearrange the equation into the form ax² + bx + c = 0
2. 2.Identify the values of a, b and c
3. 3.Substitute into x = (-b ± √(b² - 4ac)) / 2a
4. 4.Calculate both solutions, simplifying surds if needed

Worked example

Solve x² + 3x - 5 = 0 using the formula.

1. a=1, b=3, c=-5
2. x = (-3 ± √(9+20))/2 = (-3 ± √29)/2
3. x = 1.19 or x = -4.19 (2 d.p.)

Worked example

Solve 2x² - 4x - 3 = 0.

1. a=2, b=-4, c=-3
2. x = (4 ± √(16+24))/4 = (4 ± √40)/4
3. x = 2.58 or x = -0.58 (2 d.p.)

Common mistakes

• Substituting values into the formula with incorrect signs
• Forgetting to divide the whole numerator by 2a
• Making arithmetic errors calculating the discriminant b² - 4ac