GCSE MATHS • HIGHER TIER

Quadratic Formula Made Simple

Solve quadratic equations reliably by identifying a, b and c and substituting them into the quadratic formula.

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Quadratic Formula in Four Moves

1

Rearrange

Write the equation in the form ax² + bx + c = 0.

2

Identify

Find the values of a, b and c, including their signs.

3

Substitute

Put a, b and c carefully into the quadratic formula.

4

Solve

Simplify the discriminant and calculate both solutions.

What You’ll Learn

Recognise quadratic equations in the form ax² + bx + c = 0.

Identify the values of a, b and c.

Use the quadratic formula accurately.

Calculate both solutions of a quadratic equation.

Understand the discriminant b² - 4ac.

Use the discriminant to predict the number of real solutions.

Connect quadratic solutions with graph x-intercepts.

Key Rules

Standard form

Write the equation as ax² + bx + c = 0 before using the formula.

Quadratic formula

x = (-b ± √(b² − 4ac)) ÷ 2a

Identify a, b and c

a = coefficient of x², b = coefficient of x, and c = constant term. Keep the correct signs.

Two solutions

The ± means calculate using + and − to find both possible solutions.

Worked Proof: Quadratic Formula

Step-by-Step Method

  • 1. Rearrange into ax² + bx + c = 0 form.
  • 2. Identify a, b and c.
  • 3. Substitute into the formula.
  • 4. Simplify carefully.
  • 5. Calculate both solutions.

More Worked Examples

1

x² + 5x + 6 = 0

a = 1, b = 5, c = 6

x = (-5 ± √(25 - 24)) / 2
x = (-5 ± 1) / 2
Solutions: x = -2 or x = -3

2

x² - 7x + 10 = 0

a = 1, b = -7, c = 10

x = (7 ± √(49 - 40)) / 2
x = (7 ± 3) / 2
Solutions: x = 5 or x = 2

3

x² + 2x - 8 = 0

x = (-2 ± √36) / 2

x = (-2 ± 6) / 2
Solutions: x = 2 or x = -4

4

2x² + 3x - 2 = 0

a = 2, b = 3, c = -2

x = (-3 ± √25) / 4
Solutions: x = 0.5 or x = -2
  • Common GCSE Mistakes
  • Using incorrect values for a, b or c.
  • Forgetting brackets around negative numbers.
  • Making calculator-entry errors.
  • Missing one of the two solutions.
  • Making arithmetic mistakes when simplifying.
  • Exam Tips
  • Write the equation in ax² + bx + c = 0 form before starting.
  • Copy the signs of a, b and c carefully.
  • Use brackets when substituting negative values.
  • Enter the entire numerator carefully into your calculator.
  • Remember the ± gives two possible solutions when appropriate.
  • Use the discriminant to check how many real solutions to expect.

Skills Used in Quadratic Formula

Quadratic Equations

Algebraic Substitution

Square Roots

Discriminant

Calculator Skills

Quadratic Graphs

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Frequently Asked Questions

It is used to solve quadratic equations written in the form ax² + bx + c = 0.

a is the coefficient of x², b is the coefficient of x and c is the constant term.

It is especially useful when a quadratic is difficult or impossible to solve by factorising.

The discriminant is b² - 4ac, the expression inside the square root in the quadratic formula.

A positive discriminant means the quadratic has two real solutions.

It allows both possible solutions to be calculated when the quadratic has two real roots.