# How much greater is -12x^2 - 19x + 8 than -15x^2 + 17x - 18?

Apr 7, 2018

$\left(- 12 {x}^{2} - 19 x + 8\right) - \left(- 15 {x}^{2} + 17 x - 18\right)$

$= \textcolor{red}{3 {x}^{2} - 36 x + 26}$

#### Explanation:

The question "how much greater is $a$ than $b$?" can be expressed mathematically as:

$a - b = D$

where $D$ is the difference between $a$ and $b$.

The problem then is to evaluate $D$ in the expression:

$\left(- 12 {x}^{2} - 19 x + 8\right) - \left(- 15 {x}^{2} + 17 x - 18\right) = D$

First distribute the minus sign to every term in the parentheses.

$\Rightarrow - 12 {x}^{2} - 19 x + 8 - \left(- 15 {x}^{2}\right) - \left(17 x\right) - \left(- 18\right) = D$

$\Rightarrow - 12 {x}^{2} - 19 x + 8 + 15 {x}^{2} - 17 x + 18 = D$

Now group similar terms.

$\Rightarrow \left(- 12 {x}^{2} + 15 {x}^{2}\right) + \left(- 19 x - 17 x\right) + \left(8 + 18\right) = D$

$\Rightarrow \left(- 12 + 15\right) {x}^{2} + \left(- 19 - 17\right) x + \left(8 + 18\right) = D$

$\Rightarrow 3 {x}^{2} - 36 x + 26 = D$

This is our answer. If we were to substitute any value of $x$ into the two given polynomials, the difference between them would be $\textcolor{red}{3 {x}^{2} - 36 x + 26}$.

Let's check our answer to prove that it is correct.

Substitute $x = 0$

$- 12 {\left(0\right)}^{2} - 19 \left(0\right) + 8 = 8$

$- 15 {\left(0\right)}^{2} + 17 \left(0\right) - 18 = - 18$

The difference between them is

$8 - \left(- 18\right) = \textcolor{b l u e}{26}$

and our solution gives

$3 {\left(0\right)}^{2} - 36 \left(0\right) + 26 = \textcolor{b l u e}{26}$