Quadratic equations GCSE lesson: factorising and graphs

Quadratics: practice

Show your factorisation and check each root by substitution. Enter the smaller root first when there are two. The stretch is optional.

Diagnostic question: choose and explain

1. Solve (x − 4)(x + 1) = 0.
   1. x = 4 or x = −1
   2. x = −4 or x = 1
   3. x = 4 only
   4. x = 4 or x = 1

Practice questions

1. x² + 7x + 10 = 0, so x = ___ or x = ___
2. x² − x − 12 = 0, so x = ___ or x = ___
3. x² − 9 = 0, so x = ___ or x = ___
4. x² − 6x + 9 = 0 has one solution. x = ?
5. Always, sometimes or never? A quadratic curve meets the x-axis twice.
   1. Always
   2. Sometimes
   3. Never
6. You are given x² + 2x + 5 = (x + 1)² + 4. How many real solutions does x² + 2x + 5 = 0 have?
   1. None
   2. One
   3. Two
7. Make it true: y = (x + ___)(x − ___) meets the x-axis at x = −1 and x = 4.
8. The curve y = x² + bx + 12 crosses the x-axis at x = 2. So b = ___, and the other solution is x = ___
9. x² − 5x = 0, so x = ___ or x = ___
10. Stretch: 2x² + 5x − 3 = 0, so x = ___ or x = ___

Support: For question 1: (x + 5)(x + ___) = 0. The missing number multiplies with 5 to give 10 and adds to 5 to give 7. Then solve each bracket = 0 and check both values.

Quadratics: exit

One sentence is enough.

1. A friend says (x − 4)(x + 1) = 0 gives x = −4 and x = 1. Write them one sentence to help.

Exit check

Three sketches and one sentence show who sees roots as where y = 0.

Sketch three upward curves: one that meets the x-axis twice, one that touches it once, one that never meets it. Label both axes.

Quadratics: solve and sketch

For questions 1–4 and 6, factorise, solve and check each root by substitution. Sketch y equal to the left-hand side and label where it meets the x-axis. For question 5, use the square given to explain why there are no real roots, then sketch.