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# Quadratic equations

### Quadratic formula

• If the quadratic $$ax^2 + bx + c = 0$$ cannot be factorised (or if you cannot readily determine its factors), the solutions are given by the formula:
$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$
This formula may be used to obtain the solutions of any quadratic equation.
• For example: solve $$3x^2 - 15x + 17 = 9$$

Rearranging to give $$3 x^{2} - 15 x + 8 = 0$$
Using the Quadratic formula:
\begin{eqnarray*}
x &=& \frac{-b\pm\sqrt{b^2-4ac}}{2a} \\
&=& \frac{-(-15)\pm\sqrt{(-15)^2-4\times 3 \times 8}}{2\times 3} \\
&=& \frac{15 \pm \sqrt{129}}{6} \\
&\approx& 4.39 \mbox{ or } 0.61
\end{eqnarray*}

• A quadratic can have:
• 2 real solutions (roots). This happens when ($$b^2- 4ac$$) is positive.
• 1 real solution (root). This happens when ($$b^2- 4ac$$) is 0.
• 0 real solutions (roots). This happens when ($$b^2- 4ac$$) is negative.