Question

How do you factor and solve $3{x^2} + 2x = 8$ ?

Answer 1

Hint: We will, first of all, write the general quadratic equation and the formula for its roots and then compare the given equation to it to find its roots and thus the factors.

Complete step by step solution:
We are given that we are required to factor and solve $3{x^2} + 2x = 8$.
We can write this as:-
$ \Rightarrow 3{x^2} + 2x - 8 = 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}}$
Comparing the general equation $a{x^2} + bx + c = 0$ with the given equation $3{x^2} + 2x - 8 = 0$, we will then obtain the following:-
$ \Rightarrow $a = 3, b = 2 and c = - 8
Now, putting these in the formula given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, we will then obtain the following expression:-
$ \Rightarrow x = \dfrac{{ - 2 \pm \sqrt {{{(2)}^2} - 4(3)( - 8)} }}{{2(3)}}$
Simplifying the calculations in the right hand side of the above mentioned expression, we will then obtain the following expression:-
$ \Rightarrow x = \dfrac{{ - 2 \pm \sqrt {4 + 96} }}{6}$
Simplifying the calculations in the right hand side of the above mentioned expression further, we will then obtain the following expression:-
$ \Rightarrow x = \dfrac{{ - 2 \pm 10}}{6}$
Simplifying the right hand side of the above expression, we will then obtain the following expression:-
$ \Rightarrow x = \dfrac{4}{3}, - 2$

Thus, the required answer is:-
$ \Rightarrow \left( {x - \dfrac{4}{3}} \right)\left( {x + 2} \right) = 0$ which can be written as $\left( {3x - 4} \right)\left( {x + 2} \right) = 0$.

Note: The students must note that there is an alternate way to solve the same question done as follows. Here, we will use splitting the middle term method.
We can write the given equation as follows:-
$ \Rightarrow 3{x^2} - 4x + 6x - 8 = 0$
Taking x common from first two terms and 2 common from the last two terms in the left hand side of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow x\left( {3x - 4} \right) + 2\left( {3x - 4} \right) = 0$
Now, taking out (3x – 4) common in the left hand side of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow \left( {3x - 4} \right)\left( {x + 2} \right) = 0$