# How do you find the roots, real and imaginary, of y=(x-5)(-7x-2) using the quadratic formula?

Feb 11, 2018

There are two Real roots $x = + 5$ and $x = - \frac{2}{7}$

#### Explanation:

The roots are the values of $x$ for which $y = 0$
These are easy to determine from the given factored form (see bottom)
but if you need to demonstrate the use of the quadratic formula:

First you need to convert the factored form $\left(x - 5\right) \left(- 7 x - 2\right)$
into standard form (by multiplying the factors together and arranging in standard sequence):
$\textcolor{w h i t e}{\text{XXX")color(red)(} \left(- 7\right)} {x}^{2} + \textcolor{b l u e}{33} x + \textcolor{g r e e n}{10}$
then apply the quadratic formula
$\textcolor{w h i t e}{\text{XXX}} x = \frac{- \textcolor{b l u e}{b} \pm \sqrt{{\textcolor{b l u e}{b}}^{2} - 4 \textcolor{red}{a} \textcolor{g r e e n}{c}}}{2 \textcolor{red}{a}}$

Substituting corresponding values for $\textcolor{red}{a} , \textcolor{b l u e}{b} , \mathmr{and} \textcolor{g r e e n}{c}$
color(white)("XXX")x=(-color(blue)33+-sqrt(color(blue)33^2-4 * color(red)(""(-7)) * color(green)(10)))/(2 * color(red)(""(-7)))

$\textcolor{w h i t e}{\text{XXX} x} = \frac{- 33 \pm \sqrt{1369}}{- 14}$

$\textcolor{w h i t e}{\text{XXX} x} = \frac{- 33 \pm 37}{- 14}$

$\textcolor{w h i t e}{\text{XXX"x)=4/(-14)color(white)("xxxx")or color(white)("xxxx}} \frac{- 70}{- 14}$

$\textcolor{w h i t e}{\text{XXX"x)=-2/7color(white)("xxxx")orcolor(white)("xxxx}} + 5$

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As previously noted these values can be more easily derived from the original factored form, since
if $\left(x - 5\right) \left(- 7 x - 2\right) = 0$
then
{: ("either",x-5=0," or ",-7x-2=0), (,rarrx=+5,,rarrx=-2/7) :}