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 = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$$ \Rightarrow x = \dfrac{{ - 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 $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, 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 = \dfrac{{ - 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 = \dfrac{{ - 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.