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Quadratic equations
Quadratic equations can be solved by either factorisation or using a formula. Here we will look at solving quadratic equations by factorisation. A quadratic equation is like a quadratic expression as shown in the section on factorisation but a crucial difference is that it is equal to zero. So you will have the expression followed by = 0 which means it is an equation.
The following example shows how a quadratic equation can be factorised and then solved. The first bit is the simple factorisation of a quadratic expression as shown in the section on factorisation and the second bit is even easier.
x² + 6x +5 = 0
Factorise
(x + 1)(x + 5) = 0
Then notice that the product of the 2 brackets equals 0 so either:
The first bracket equates to 0.
x + 1 = 0 so x = -1 because 0 x (x + 5) = 0
or
The second bracket equates to 0.
x + 5 = 0 so x = -5 because (x + 1) x 0 = 0
Solutions: x = -1, x = -5
Usually there are two solutions as in the above example but sometimes there can be only one. The solutions or roots are the 2 points at which the curve crosses the x-axis giving 2 solutions. The parabola may only touch the x-axis at one point resulting in 1 solution (or two equal roots).
Parabola crosses x-axis at 2 points
Parabola crosses the x-axis at 2 points resulting in 2 solutions or roots
Parabola touches the x-axis at 1 point
Parabola touches the x-axis at 1 point giving one solution or 2 equal roots
Exercise 1
Solve the following quadratic equations using factorisation.
1. x² + 8x + 12 = 0
2. x² - 3x - 4 = 0
3. x² + 11x + 28 = 0
4. 2x² + 5x - 3 = 0
5. 4x² - 11x + 6 = 0
6. x² + 8x + 12 = 0
7. x² + 2x - 35 = 0
8. 6x² - 11y - 7 = 0
9. 12x² - 7x - 12 = 0
10. 3x² + 5x - 2 = 0