# How do you solve 3r ^ { 2} - 8r - 16= 0?

Jan 18, 2018

$r = - 4 , r = - \frac{4}{3}$

#### Explanation:

This is a factoring problem.

You start by multiplying $3$ by $- 16$, which gives you $- 48$.

Next, try to find two numbers that multiply to be $- 48$ and add to be $- 8$.

Those numbers are $4$ and $- 12$
($4 - 12 = - 8$) and ($4 \cdot - 12 = - 48$)

Then, split the middle term in the equation, using those numbers.
$3 {r}^{2} + 4 r - 12 r - 16 = 0$

Look at the first two terms, and pull out what you can.
$r \left(3 r + 4\right)$
Then look at the next two terms, and do the same.
$4 \left(3 r + 4\right)$

The full equation is now:
$r \left(3 r + 4\right) + 4 \left(3 r + 4\right) = 0$

Because $3 r + 4$ is common in both terms, you can "pull it out".
$\left(3 r + 4\right) \left(r + 4\right) = 0$

Then, you set each factor equal to zero and solve.
$3 r + 4 = 0$
$3 r + 4 - 4 = 0 - 4$
$3 r = - 4$
$r = - \frac{4}{3}$

$r + 4 = 0$
$r + 4 - 4 = 0 - 4$
$r = - 4$

Jan 18, 2018

$r$ equals $- \frac{4}{3}$ and $4$

#### Explanation:

First let's see what we can do about the coefficients. Are there any common factors between $3 , 8 , 16$? No there aren't since $3$ is prime and the other two aren't multiples of $3$.

Now we need to factor. When the coefficient is not equal to $1$ we have to use a method that isn't as straight forward. Some people like to use the box method but I prefer to use an alternative method called factor by grouping

First find the factor:

$3 {r}^{2} \textcolor{\pi n k}{- 8} r - 16$

$\textcolor{w h i t e}{.} + \textcolor{w h i t e}{.} 8$
$\textcolor{w h i t e}{.} \times \textcolor{w h i t e}{.} 48$
....................
$\textcolor{w h i t e}{.} 1 \textcolor{w h i t e}{. . -} 48$
$\textcolor{w h i t e}{.} 2 \textcolor{w h i t e}{. . -} 24$
$\textcolor{w h i t e}{.} 3 \textcolor{w h i t e}{. . -} 16$
color(red)(color(white)(.) 4 color(white)(..) -12 $= \textcolor{\pi n k}{- 8}$

Now we fill the blanks with the factors. It doesn't matter which one goes where, just make sure to multiply them by $r$:

$\left(\textcolor{b l u e}{3 {r}^{2}} + \textcolor{w h i t e}{- 12 r}\right) + \left(\textcolor{w h i t e}{4 r} - \textcolor{b l u e}{16}\right)$

$\left(\textcolor{b l u e}{3 {r}^{2}} + \textcolor{red}{- 12 r}\right) + \left(\textcolor{red}{4 r} - \textcolor{b l u e}{16}\right)$

Factor out the greatest common factor:

$3 r \left(r - 4\right) + 4 \left(r - 4\right)$

The parentheses have the same value, so we're good. Factor $\left(r - 4\right)$ from the expression:

$\left(r - 4\right) \left(3 r + 4\right)$

Now we have our final factored form. but we still haven't solved the problem. Set each factor equal to $0$ and solve:

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

$r - 4 = 0$

$r = 4$

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

$3 r + 4 = 0$

$3 r = - 4$

$r = - \frac{4}{3}$

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

So, $r$ equals $- \frac{4}{3}$ and $4$