Question

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

Answer 1

Answer:

$a=3x-1$

Step-by-step explanation:

Given:

$a=\dfrac{(x+2)^2-(2x-3)^2}{-x+5}$

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:

$a=\dfrac{(x-5)(-3x+1)}{-x+5}\\ \\a=\dfrac{(x-5)(-3x+1)}{-(x-5)}\\ \\a=\dfrac{-3x+1}{-1}\\ \\a=3x-1$