Question

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

Answer 1

`(-12x^2-19x+8)-(-15x^2+17x-18)`

`=color(red)(3x^2-36x+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:

`(-12x^2-19x+8)-(-15x^2+17x-18)=D`

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

`rArr-12x^2-19x+8-(-15x^2)-(17x)-(-18)=D`

`rArr-12x^2-19x+8+15x^2-17x+18=D`

Now group similar terms.

`rArr(-12x^2+15x^2)+(-19x-17x)+(8+18)=D`

`rArr(-12+15)x^2+(-19-17)x+(8+18)=D`

`rArr3x^2-36x+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 `color(red)(3x^2-36x+26)`.

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

Substitute `x=0`

`-12(0)^2-19(0)+8 = 8`

`-15(0)^2+17(0)-18 = -18`

The difference between them is

`8-(-18)=color(blue)26`

and our solution gives

`3(0)^2-36(0)+26 = color(blue)26`