Question

How do you factor $x^2+11x=180$?

Answer 1

Hint: We will bring the constant on the left hand side and we will get a quadratic equation in $x$. Then, we will use the quadratics formula to solve that and thus find the factors related to the roots we obtained.

Complete step by step solution:
We are given that we are required to factor $x^2+11x=180$.
Taking 180 from addition in the right hand side to subtraction in the left hand side, we will then obtain the following equation with us:-
$\Rightarrow x^2+11x-180=0$
We know that the general quadratic equation is given by $a{x}^2+bx+c=0$, where a, b and c are constants.
Now, we know that its roots are given by the following expression:-
$\Rightarrow x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$$\Rightarrow x={\displaystyle \frac{-11\pm \sqrt{{(11)}^2-4(1)(-180)}}{2(1)}}$
Comparing the general equation $a{x}^2+bx+c=0$ with the given equation $x^2+11x-180=0$, we will then obtain the following:-
$\Rightarrow$a = 1, b = 11 and c = - 180
Now, putting these in the formula given by $x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$, we will then obtain the following expression:-
Simplifying the calculations in the right hand side of the above mentioned expression, we will then obtain the following expression:-
$\Rightarrow x={\displaystyle \frac{-11\pm \sqrt{121+720}}{2}}$
Simplifying the calculations in the right hand side of the above mentioned expression further, we will then obtain the following expression:-
$\Rightarrow x={\displaystyle \frac{-11\pm 29}{2}}$
Simplifying the calculations on the right hand side, we will then obtain the following possible values of x:-
$\Rightarrow x=9,-20$
Thus, the factors of $x^2+11x-180=0$ are $\left(x-9\right)$ and $\left(x+20\right)$.

Hence, we have: $x^2+11x=180\Rightarrow \left(x-9\right)\left(x+20\right)=0$

Note:
The students must note that there is an alternate way to solve the same question.
Alternate way:
We will use the method of splitting the middle term.
We are given that we are required to factor $x^2+11x=180$.
Taking 180 from addition in the right hand side to subtraction in the left hand side, we will then obtain the following equation with us:-
$\Rightarrow x^2+11x-180=0$
We can write this equation as follows:-
$\Rightarrow x^2-9x+20x-180=0$
Taking x common from first two terms and 20 common from last two terms in the left hand side of the above equation, we will then obtain the following equation with us:-
$\Rightarrow x\left(x-9\right)+20\left(x-9\right)=0$
Taking $\left(x-9\right)$ common from it, we will then obtain:-
$\Rightarrow \left(x-9\right)\left(x+20\right)=0$
Thus, we have the required answer.