# Two sides of a triangle are (6x^2-12x) feet and (x^2+30x-15) feet. What is the length of the third side if the perimeter is 14x^2+34x+15?

Dec 24, 2017

The length of the third side of the triangle is $7 {x}^{2} + 16 x + 30$.

#### Explanation:

We know that the perimeter is $14 {x}^{2} + 34 x + 15$.

We also know that two sides of the triangle are $6 {x}^{2} - 12 x$ and ${x}^{2} + 30 x - 15$.

To find the third side of the triangle, we must subtract the length of the two sides from the perimeter:
$\left(14 {x}^{2} + 34 x + 15\right) - \left(6 {x}^{2} - 12 x\right) - \left({x}^{2} + 30 x - 15\right)$

Notice that the minus applies to everything inside its parenthesis.
$14 {x}^{2} + 34 x + 15 - 6 {x}^{2} + 12 x - {x}^{2} - 30 x + 15$

Now we need to combine like terms:
$14 {x}^{2}$, $- 6 {x}^{2}$, and $- {x}^{2}$ are like terms (all contain ${x}^{2}$), so...
$14 {x}^{2} - 6 {x}^{2} - {x}^{2} = 7 {x}^{2}$

$34 x$, $12 x$, and $- 30 x$ are like terms (all contain $x$), so...
$34 x + 12 x - 30 x = 16 x$

$15$ and $15$ are in like terms, so...
$15 + 15 = 30$

Finally, let's combine all these together to form one expression:
$7 {x}^{2} + 16 x + 30$

If we want to check our answer to make sure we did it correctly, we just add up the three sides and set it equal to the perimeter. If they equal, then we did it correctly, if not, then we have to check/do our work again.
$6 {x}^{2} - 12 x + {x}^{2} + 30 x - 15 + 7 {x}^{2} + 16 x + 30 = 14 {x}^{2} + 34 x + 15$
$14 {x}^{2} + 34 x + 15 = 14 {x}^{2} + 34 x + 15$

The length of the third side of the triangle is $7 {x}^{2} + 16 x + 30$.