Finding roots by factorising

If a quadratic equation can be factorised, the factors can be used to find the roots of the equation.

Example

x^2 + x - 6 = 0

The equation factorises to give (x + 3) (x - 2) = 0 so the solutions to the equation x^2 + x - 6 = 0 are x = -3 and  x = 2.

The graph of y = x^2 + x - 6 crosses the x-axis at x = -3 and x = 2.

The graph of y = x^2 + x - 6 crosses the x-axis at x = -3 and x = 2.

Example

x^2 - 6x + 9 = 0

The equation factorises to give (x – 3)(x – 3) = 0 so there is just one solution to the equation,  x = 3.

The graph of y = x^2 - 6x + 9 touches the x-axis at  x = 3.

The graph of y = x^2 - 6x + 9  touches the x-axis at x = 3

Example

x^2 + 2x + 5 = 0.

Using the quadratic formula to try to solve this equation, a = 1, b = 2 and c = 5 which gives:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x = \frac{-2 \pm \sqrt{2^2 - 4 \times 1 \times 5}}{2} = \frac{-2\pm \sqrt{-16}}{2}.

It is not possible to find the square root of a negative number, so the equation has no solutions.

The graph of y = x^2 + 2x + 5 does not cross or touch the x-axis so the equation x^2 + 2x + 5 = 0 has no roots.

The graph of y = x^2 + 2x + 5 doesn't cross or touch the x-axis as the equation x^2 + 2x + 5 = 0 has no roots.

Finding the y-intercept

The graph of the quadratic equation y = ax^2 + bx + c crosses the y-axis at the point (0, c). The x-coordinate of any point on the y-axis has the value of 0 and substituting x = 0 into the equation y = ax^2 + bx + c gives y = c.

Question

Find the y-intercept of the following quadratic functions:

a)  y = x^2 + 3x – 2

b)  y = x^2 + 17

c)  y = x^2 + 5x

a) The constant term is -2, so the y-intercept is (0, -2)

b) The constant term is 17, so the y-intercept is (0, 17)

c) The constant term is 0, so the y-intercept is (0, 0)