Q 116. - Intermediate Algebra

Question

Factor Trinomials of the Form $a{x}^2+bx+c$ using the ‘ac’ Method.

$6p^2+p-22$

Step-by-Step Solution

Answer

The solution is $\begin{array}{rcl}& & (p+2)(6p-11)\end{array}$.

Step 1. Given information

The given trinomial is $6p^2+p-22$.

Step 2. Find greatest common factor.

There is no greatest common factor.

Step 3. Compare the given trinomial to a x 2 + b x + c and then find the product of a c .

The values are as follows:

$\begin{array}{rcl}a& =& 6\\ b& =& 1\\ c& =& -22\end{array}$

The product is as follows:

$\begin{array}{rcl}ac& =& 6(-22)\\ & =& -132\end{array}$

Step 4. Determine two numbers that multiply to a c and add to b .

Two numbers that multiply to $ac$ and add to $b$ are as follows:

$\begin{array}{rcl}12(-11)& =& -132\\ & =& ac\\ 12-11& =& 1\\ & =& b\end{array}$

So, two numbers are $12$ and $-11$.

Now, split the middle term of the given trinomial $6p^2+p-22$ as follows:

$6p^2+p-22=6p^2+12p-11p-22$

Step 5. Factor by grouping.

$\begin{array}{rcl}6p^2+p-22& =& 6p(p+2)-11(p+2)\\ & =& (p+2)(6p-11)\end{array}$

Step 6. Check by multiplying ( p + 2 ) ( 6 p - 11 ) .

$\begin{array}{rcl}(p+2)(6p-11)& =& 6p^2+12p-11p-22\\ & =& 6p^2+p-22\end{array}$

Hence, the factor is $\begin{array}{rcl}& & (p+2)(6p-11)\end{array}$.