08Bquadraticformula

= The Quadratic Formula =

When solving any quadratic equation in the form:

math . \qquad \qquad ax^2 + bx + c = 0 \qquad \qquad \big\{ \text{where a, b, c are any numbers} \big\} math

we can use the ** quadratic formula **:

This formula can be obtained using completing the square, see here:

** Example 1 **
Solve the following quadratic equations
 * 1) 3x² – 10x – 8 = 0
 * 2) –2x² + 6x + 2 = 0


 * Solution: **

1. Solve 3x² – 10x – 8 = 0

... ... ... a = 3, ... b = –10, ... c = –8

math \\ . \qquad x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \\. \\ . \qquad x = \dfrac{10 \pm \sqrt{10^2-4 \times 3 \times -8}}{2 \times 3} \\. \\ . \qquad x = \dfrac{10 \pm \sqrt{100 + 96}}{6} \\. \\ . \qquad x = \dfrac{10 \pm \sqrt{196}}{6} math . math \\ . \qquad x = \dfrac{10 \pm 14}{6} \\. \\ . \qquad x = \dfrac{10 + 14}{6} \quad \text{or} \quad x = \dfrac{10 - 14}{6} \\. \\ . \qquad \;\; x = \dfrac{24}{6} \qquad \;\; \text{or} \quad \;\; x = \dfrac{-4}{6} \\. \\ . \qquad \quad x = 4 \qquad \quad \text{or} \quad \;\; x = \dfrac{-2}{3} math

2. Solve –2x² + 6x + 2 = 0

... ... ... a = –2, ... b = 6, ... c = 2

math \\ . \qquad x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \\. \\ . \qquad x = \dfrac{-6 \pm \sqrt{(-6)^2-4 \times -2 \times 2}}{2 \times -2} \\. \\ . \qquad x = \dfrac{-6 \pm \sqrt{36 + 16}}{-4} \\. \\ . \qquad x = \dfrac{-6 \pm \sqrt{52}}{-4} math . math \\ . \qquad x = \dfrac{-6 \pm 2\sqrt{13}}{-4} \\. \\ . \qquad x = \dfrac{3 \pm \sqrt{13}}{2} math

No Solutions

 * When using the quadratic formula,
 * if the square root contains a __negative__ value,
 * there will be ** no real solutions **.

** Example 2 **
Solve the following quadratic equation
 * 1) x² + 2x + 6 = 0


 * Solution: **

1. Solve x² + 2x + 6 = 0

... ... ... a = 1, ... b = 2, ... c = 6

math \\ . \qquad x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \\. \\ . \qquad x = \dfrac{-2 \pm \sqrt{2^2-4 \times 1 \times 6}}{2 \times 1} \\. \\ . \qquad x = \dfrac{-2 \pm \sqrt{4 - 24}}{2} \\. \\ . \qquad x = \dfrac{-2 \pm \sqrt{-20}}{2} math


 * Square root contains a negative value
 * Hence this equation has ** no real solutions **.