Question

How do you solve the quadratic equation $3{x^2} + 9x = 12$?

Answer 1

Hint: First bring the 12 from the RHS to the LHS and then take out 3 common and cross it off. Then, use the method of “splitting the middle term” to make the factors and thus we have the roots.

Complete step-by-step solution:
We are given that we are required to solve $3{x^2} + 9x = 12$.
Taking 12 from addition in the right hand side to subtraction in the left hand side of the above mentioned equation, we will then obtain the following equation with us:-
$ \Rightarrow 3{x^2} + 9x - 12 = 0$
Taking 3 common and crossing it off from the left hand side of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow {x^2} + 3x - 4 = 0$
We can write the above mentioned equation as follows:-
$ \Rightarrow {x^2} - x + 4x - 4 = 0$
Taking x common from the first two terms from the above mentioned equation, we will then obtain the following equation with us:-
$ \Rightarrow x\left( {x - 1} \right) + 4x - 4 = 0$
Taking 4 common from the last two terms from the above mentioned equation, we will then obtain the following equation with us:-
$ \Rightarrow x\left( {x - 1} \right) + 4\left( {x - 1} \right) = 0$
Taking (x – 1) common from the last two terms in the latter factor, we will then obtain the following equation with us:-
$ \Rightarrow \left( {x - 1} \right)\left( {x + 4} \right) = 0$
Thus, we have the required roots as 1 and – 4.

Note: The students must notice that we have an alternate way of factoring the quadratic equation involved in it as well. The alternate way is as follows:-
The given equation is ${x^2} + 3x - 4 = 0$.
Using the quadratic formula given by if the equation is given by $a{x^2} + bx + c = 0$, its roots are given by the following equation:-
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Thus, we have the roots of ${x^2} + 3x - 4 = 0$ given by:
$ \Rightarrow x = \dfrac{{ - 3 \pm \sqrt {{{(3)}^2} - 4 \times ( - 4)} }}{2}$
Simplifying the calculations in the square root in the numerator of the right hand side, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{{ - 3 \pm \sqrt {9 + 16} }}{2}$
Simplifying the calculations in the square root in the numerator of the right hand side further, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{{ - 3 \pm 5}}{2}$
Hence, the roots are 1 and - 4.