Year 9: Quadratic equations

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1. What is a quadratic equation?

A quadratic equation is an equation that can be written in the form:

ax² + bx + c = 0

Where a, b, and c are constants, and a ≠ 0.

2. The shape of a quadratic graph

The graph of a quadratic equation is a curve called a parabola. It can open upwards or downwards depending on the sign of a.

• If a is positive, the parabola opens upwards.
• If a is negative, the parabola opens downwards.

3. Solving quadratic equations by factorising

Many quadratic equations can be solved by factorising the expression into two brackets:

ax² + bx + c = (x + m)(x + n) = 0

Then set each bracket equal to 0 and solve for x:

x + m = 0 or x + n = 0

Example: Solve x² − 5x + 6 = 0
Factor: (x − 2)(x − 3) = 0 ⇒ x = 2 or x = 3

4. Using the quadratic formula

If the equation cannot be factorised easily, use the quadratic formula:

x = (−b ± √(b² − 4ac)) ÷ 2a

Example: Solve x² − 4x − 5 = 0
a = 1, b = −4, c = −5
x = (4 ± √(16 + 20)) ÷ 2 = (4 ± √36) ÷ 2 = (4 ± 6) ÷ 2
Solutions: x = 5 or x = −1

5. Discriminant and nature of roots

The part under the square root in the quadratic formula is called the discriminant: D = b² − 4ac

• If D > 0 → two real solutions
• If D = 0 → one real solution
• If D < 0 → no real solutions

6. Practice Questions

1. Solve x² − 7x + 12 = 0 by factorising.
2. Solve x² + 5x + 6 = 0 using the quadratic formula.
3. Determine the number of solutions for x² + 4x + 5 = 0.

7. Solutions

1. (x − 3)(x − 4) = 0 ⇒ x = 3 or x = 4
2. a = 1, b = 5, c = 6
   D = 25 − 24 = 1
   x = (−5 ± 1) ÷ 2 ⇒ x = −2 or x = −3
3. D = 16 − 20 = −4 ⇒ no real solutions