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Quadratic Factorisation Techniques

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Quadratic Factorisation Techniques

Factorise where asked. Show your working for questions that involve completing the square or solving an equation.

Review: Expanding and Factors

1.Expand and simplify: (x + 4)(x + 3).

2.Choose the factorisation of x² + 8x + 15.

(x + 3)(x + 5)
(x + 1)(x + 15)
(x + 2)(x + 7)
(x - 3)(x - 5)

3.Find two integer factors of 18 whose sum is 9.

Factorising Quadratics

4.Factorise fully: x² + 11x + 24.

5.Factorise fully: x² + 3x - 18.

6.Factorise fully: x² - 10x + 21.

7.Factorise fully: 2x² + 7x + 3.

8.Match each quadratic expression to its factorised form.

x² + 6x + 8
x² - 3x - 4
x² - 8x + 15
x² + x - 12

(x - 2)(x + 6)
(x - 4)(x + 1)
(x + 2)(x + 4)
(x - 5)(x - 3)

Completing the Square and Solving

9.Rewrite x² + 10x + 6 in completed-square form. Show how you split the middle term.

10.Use the quadratic formula to solve 2x² - 3x - 2 = 0. Give both solutions.

11.For x² - 6x + 13 = 0, what does the discriminant show about the real solutions?

There are two distinct real solutions.
There is one repeated real solution.
There are no real solutions.
The equation has infinitely many real solutions.

Applications

12.A quadratic has zeros x = -4 and x = 2, and its leading coefficient is 1. Write its rule in factorised form.

13.A toy rocket's height is h = -5t² + 25t + 30 metres, where t is the time in seconds after launch. Factorise the expression and find when the rocket reaches the ground.

14.In your own words, explain why setting each factor to zero can give the x-intercepts of a quadratic graph.