Question

How do you factor $18{{x}^{2}}+3x-10$ by grouping?

Answer 1

Hint: In the above problem, the expression is the quadratic expression because the degree of this expression is 2. We are going to factorize the given quadratic equation by multiplying the coefficient of ${{x}^{2}}$ and the constant. Then we are going to find the factors of this multiplication and we will add or subtract these factors in such a fashion so that we will get the coefficient of x and then replace the coefficient of x with these factors. And then we can easily factorize.

Complete step-by-step answer:
The expression given above is as follows:
$18{{x}^{2}}+3x-10$
The coefficient of ${{x}^{2}}$ is 18 in the above and constant is -10 so multiplying 18 by 10 we get 180 so factoring 180 we get,
Factorization of 180 is as follows:
$\begin{align}
  & 180=1\times 180 \\
 & 180=2\times 90 \\
 & 180=3\times 60 \\
 & 180=4\times 45 \\
 & 180=5\times 36 \\
 & 180=6\times 30 \\
 & 180=9\times 20 \\
 & 180=10\times 18 \\
 & 180=15\times 12 \\
\end{align}$
Now, we are going to find the factors and addition or subtraction of them will give us the coefficient of x. As you can take the last factor which is (15 and 12) and on subtracting 12 from 15 we will get the coefficient of x so substituting $\left( 15-12 \right)$ in place of 3 in the above quadratic expression we get,
$\begin{align}
  & \Rightarrow 18{{x}^{2}}+\left( 15-12 \right)x-10 \\
 & \Rightarrow 18{{x}^{2}}+15x-12x-10 \\
\end{align}$
Taking 3x as common from the first two terms and -2 from the last two terms we get,
$\Rightarrow 3x\left( 6x+5 \right)-2\left( 6x+5 \right)$
Now, taking $6x+5$ as common from the above expression we get,
$\Rightarrow \left( 3x-2 \right)\left( 6x+5 \right)$
Hence, we have factorized the given quadratic expression and the factors are $\left( 3x-2 \right)\left( 6x+5 \right)$.

Note: You can check whether the factors found are correct or not by multiplying the two factors and see whether we are getting the same answer or not.
The factors which are getting above are as follows:
$\left( 3x-2 \right)\left( 6x+5 \right)$
Multiplying the two brackets to each other we get,
$\begin{align}
  & 3x\left( 6x \right)+3x\left( 5 \right)-2\left( 6x \right)-10 \\
 & =18{{x}^{2}}+15x-12x-10 \\
 & =18{{x}^{2}}+3x-10 \\
\end{align}$
As you can see that we are getting the same quadratic expression which we started with so the factors which we found in the above solution are correct.