Question

Factor the trinomial

$5x^2-7x-6$

Short answer

The solution $5x^2-7x-6=\left(5x+3\right)\left(x-2\right)$

Step-by-step solution

Step 1. Apply the concept of factoring trinomials.

To factor a trinomial of the form $a{x}^2+bx+c;a\ne 0$, we note that

$a{x}^2+bx+c=\left(px+r\right)\left(qx+s\right)=pq{x}^2+\left(ps+qr\right)x+rs$

So, we need to choose numbers p, q, r and s so that $pq=a$, $ps+qr=b$and $rs=c$.

Step 2. Analyze the trinomial.

Given$5x^2-7x-6$

This is of the form $a{x}^2+bx+c$where $a=5,b=-7,c=-6$

The factorization of the trinomial will be $\left(px+r\right)\left(qx+s\right)=pq{x}^2+\left(ps+qr\right)x+rs$

So, we need to choose numbers r and s so that $pq=a$, $ps+qr=b$and $rs=c$.

Step 3. Find the integers to factor the trinomial.

We need to choose integers r and s so that $pq=5,ps+qr=-7,rs=-6$

By trial and error, we find that the two integers are $p=5,q=1,r=3,s=-2$

$pq=5\cdot 1=5$

$ps+qr=-10+3=-7$

$rs=3\cdot (-2)=-6$

Step 4. Factor the trinomial.

The factorization of the trinomial is

$\begin{array}{l}5x^2-7x-6=\left(px+r\right)\left(qx+s\right)\\ \Rightarrow 5x^2-7x-6=\left(5x+3\right)\left(x-2\right)\end{array}$