Question

How do you solve $2{x^3} - 8x = 0$ by factoring?

Answer 1

Hint: We will first take out x common from the equation and thus we get a quadratic equation in between and now we will find the roots of that quadratic and thus we have factorization.

Complete step-by-step answer:
We are given that we need to solve the equation $2{x^3} - 8x = 0$ by factoring.
Now, since both the term have 2x, we can take 2x common and write the given equation as the following:-
$ \Rightarrow 2x\left( {{x^2} - 4} \right) = 0$ …………..(1)
Now, we know that we have formula or an identity given by the following expression:-
$ \Rightarrow {a^2} - {b^2} = (a - b)(a + b)$
Replacing a by x and b by 2 in the above equation, we will then get:-
$ \Rightarrow {x^2} - 4 = (x - 2)(x + 2)$
Putting this in equation number (1), we will get:-
$ \Rightarrow 2x\left( {x - 2} \right)\left( {x + 2} \right) = 0$
Now, we have got three possibilities, either 2x = 0 or (x – 2) = 0 or (x + 2) = 0.
This implies that either x = 0 or x = 2 or x = - 2.

Hence, we have the roots of $2{x^3} - 8x = 0$ as 0, -2 and 2.

Note:
The students must note that there is an alternate way to do the same.
Let us do that:-
We are given that we need to solve the equation $2{x^3} - 8x = 0$ by factoring.
Now, since both the term have 2x, we can take 2x common and write the given equation as the following:-
$ \Rightarrow 2x\left( {{x^2} - 4} \right) = 0$ …………..(1)
The general quadratic equation is of the form: $a{x^2} + bx + c = 0$ and its roots are given by the formulas: $x = \dfrac{{ - b \pm \sqrt D }}{{2a}}$ where D is the discriminant and is given by $D = {b^2} - 4ac$.
If we compare this to ${x^2} - 4$, we have a = 1, b = 0 and c = - 4.
Now, to get the roots, let us find the roots by putting the above mentioned values in the formulas mentioned above. So, we will get the discriminant as $D = {\left( 0 \right)^2} - 4 \times 1 \times \left( { - 4} \right)$.
Simplifying the calculation by opening the required square on the right hand side to obtain the following:-
$ \Rightarrow D = 0 - 4 \times 1 \times \left( { - 4} \right)$
Simplifying the calculations further to obtain the following expression:-
$ \Rightarrow $D = - ( - 16)
Simplifying it further in the right hand side to obtain:-
$ \Rightarrow $D = 16
Now, let us put this in the formula: $x = \dfrac{{ - b \pm \sqrt D }}{{2a}}$, we will then get:-
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {16} }}{{2a}}$
Now putting a = 1 and b = 0 in the above expression, we will then obtain expression:-
$ \Rightarrow x = \dfrac{{ - 0 \pm \sqrt {16} }}{{2\left( 1 \right)}}$
Simplifying the calculations in the above expression, we will then obtain:-
$ \Rightarrow x = \dfrac{{ \pm \sqrt {16} }}{2}$
Hence, the roots are $x = 2$ and $x = - 2$.
Thus, we have $ \Rightarrow 2{x^3} - 8x = x(x - 2)(x + 2) = 0$