How do you find the roots for x^2 – 14x – 32 = 0?

Jun 6, 2015

In an equation of the following form

$a {x}^{2} + b x + c = 0$

the method to find the roots is:

1) calculate $\Delta = {b}^{2} - 4 a c$
2) if $\Delta = 0$ there is only one root ${x}_{0} = \frac{- b}{2 a}$
3) if $\Delta > 0$ there are two roots ${x}_{-} = \frac{- b - \sqrt{\Delta}}{2 a}$
and ${x}_{+} = \frac{- b + \sqrt{\Delta}}{2 a}$
4) if $\Delta < 0$ there is no real solution

Example:

${x}^{2} - 14 x - 32 = 0$

rarr a=1; b=-14; c=-32

$\rightarrow \Delta = {\left(- 14\right)}^{2} - 4 \cdot 1 \cdot \left(- 32\right) = 196 + 128 = 324$

$\Delta > 0$ therefore we have two roots:

${x}_{-} = \frac{14 - \sqrt{324}}{2} = \frac{14 - 18}{2} = - \frac{4}{2} = - 2$

${x}_{+} = \frac{14 + \sqrt{324}}{2} = \frac{14 + 18}{2} = \frac{32}{2} = 16$

Let us check the validity of our results:

${\left(- 2\right)}^{2} - 14 \cdot \left(- 2\right) - 32 = 4 + 28 - 32 = 0 \rightarrow O K$

${\left(16\right)}^{2} - 14 \cdot \left(16\right) - 32 = 256 - 224 - 32 = 0 \rightarrow O K$

Jun 6, 2015

There are several methods we could use. Here's one.

Notice that $2 \cdot 16 = 32$ and the difference between 2 and 16 is 14.

So, if the signs work out, we can factor.

${x}^{2} - 14 x - 32 = \left(x + 2\right) \left(x - 16\right)$

So, ${x}^{2} - 14 x - 32 = 0$ if and only if

$\left(x + 2\right) \left(x - 16\right) = 0$

Thus, we need

$x + 2 = 0$ or $x - 16 = 0$

The solutions are:

$x = - 2$, $x = 16$.