Question

1. the base of a triangle is $\frac{2x^{2}+5x - 3}{x^{2}-x - 12}$ and the height is $\frac{x^{2}-x - 2}{2x^{2}-5x + 2}$

Explanation:

Step1: Recall the area formula of a triangle

The area formula of a triangle is $A=\frac{1}{2}\times base\times height$. Here, $base = \frac{2x^{2}+5x - 3}{x^{2}-x - 12}$ and $height=\frac{x^{2}-x - 2}{2x^{2}-5x + 2}$.

Step2: Factor the polynomials

Factor $2x^{2}+5x - 3=(2x - 1)(x + 3)$; $x^{2}-x - 12=(x - 4)(x+3)$; $x^{2}-x - 2=(x - 2)(x + 1)$; $2x^{2}-5x + 2=(2x - 1)(x - 2)$.

Step3: Substitute the factored - forms into the area formula

$A=\frac{1}{2}\times\frac{(2x - 1)(x + 3)}{(x - 4)(x + 3)}\times\frac{(x - 2)(x + 1)}{(2x - 1)(x - 2)}$.

Step4: Cancel out the common factors

Cancel out the common factors $(2x - 1)$, $(x + 3)$ and $(x - 2)$ in the numerator and denominator. We get $A=\frac{1}{2}\times\frac{1}{x - 4}\times\frac{x + 1}{1}=\frac{x + 1}{2(x - 4)}$, where $x
eq - 3,x
eq\frac{1}{2},x
eq2,x
eq4$.

Answer:

$\frac{x + 1}{2(x - 4)}$