Question

How do you factor ${{x}^{2}}+13x+12$ ?

Answer 1

Hint: We are given ${{x}^{2}}+13x+12$ , we are asked to find the factor form of this, to do so we will first understand the type of equation we have, once we get that we will find the greatest common factor from each term then in the remaining terms we find factors using the middle term.
We use $a\times c$ in such a way that its sum or difference form the ‘b’ of the equation $a{{x}^{2}}+bx+c$ .
Once we split that we will unite all factors of the equation and we get our required answer.

Complete step-by-step solution:
We are given ${{x}^{2}}+13x+12$ ,we are asked to find the factor of it.
To find the factor of the equation, we should see that as the highest power is 2 so it is 2 degree polynomial. So it is a quadratic equation.
Now, to find the factor, we will first find the possible greatest common factor of all these. In 1, 13 and 12, we can see that there is no common term to each of these terms so we cannot separate anything out of this
Now we will use the middle term to split
In middle term split apply on $a{{x}^{2}}+bx+c$ we produce ‘a’ by ‘c’ and then factor ‘ac’ in such a way that
if the product is ‘ac’ while the sum or difference is made up to ‘b’.
Now, we have middle term split on ${{x}^{2}}+13x+12$
We have a=1 b=13 and c=12
So, we use these values to find two terms which help us in splitting the middle term .
Now we can see that $12\times 1=12$ and $12+1=13$
So we use this to split the middle term.
So,
${{x}^{2}}+13x+12$ become
${{x}^{2}}+\left( 12+1 \right)x+12$
Opening brackets
${{x}^{2}}+12x+x+12$
We take common in the first 2 terms and the last 2 terms. So we get –
$x\left( x+12 \right)+1\left( x+12 \right)$
As $x+12$ is same, so we get –
$\left( x+1 \right)\left( x+12 \right)$
So, we get –
${{x}^{2}}+13x+12=\left( x+1 \right)\left( x+12 \right)$
So, factor form of ${{x}^{2}}+13x+12$ is $\left( x+1 \right)\left( x+12 \right)$

Note: While we find the middle term using factor $a\times c$ , we need to keep in mind that when the sign of ‘a’ and ‘c’ are same then ‘b’ is obtained by addition only, if the sign of ‘a’ and ‘c’ are different then ‘b’ can be obtained using only subtraction.
So, as we have $a=1$ and $c=12$ have the same sign so ‘b’ is obtained as $1+12=13$ by addition of ‘1’ and ‘12’.
We can always cross check that –
Product of
 $\begin{align}
  & \left( x+1 \right)\left( x+12 \right)={{x}^{2}}+12x+x+12 \\
 & ={{x}^{2}}+13x+12 \\
\end{align}$
So our factors are correct.