Question

`a = \frac{ {(x + 2)}^(2) - {(2x - 3)}^(2) }{ - x + 5}`
make a in the simplest form

Answer 1

`a=3x-1`

Explanation:

Given:

`a=((x+2)^2-(2x-3)^2)/(-x+5)`

Use formulas

`(u+v)^2=u^2+2uv+v^2\\ \\(u-v)^2=u^2-2uv+v^2`

Therefore,

`(x+2)^2=x^2+2\cdot x\cdot 2+2^2=x^2+4x+4\\ \\(2x-3)^2=(2x)^2-2\cdot (2x)\cdot 3+3^2=4x^2-12x+9`

Now

`(x+2)^2-(2x-3)^2\\ \\=x^2+4x+4-(4x^2-12x+9)\\ \\=x^2+4x+4-4x^2+12x-9\\ \\=-3x^2+16x-5`

Factor it:

`-3x^2+16x-5\\ \\=-3x^2+15x+x-5\\ \\=-3x(x-5)+(x-5)\\ \\=(x-5)(-3x+1)`

Hence, the fraction can be simplified as follows:

`a=((x-5)(-3x+1))/(-x+5)\\ \\a=((x-5)(-3x+1))/(-(x-5))\\ \\a=(-3x+1)/(-1)\\ \\a=3x-1`