Question

How do you solve $5{{x}^{2}}=100$ using factoring?

Answer 1

Hint: In this question, we have to find the value of x. Since the question itself mentions using factoring, therefore we will use the factoring method. Thus, we start solving this problem by dividing 5 on both sides of the equation. Then on further simplification, we will subtract 20 on both sides of the equation and then apply the algebraic identity $(a+b)(a-b)={{a}^{2}}-{{b}^{2}}$ in the equation. On further simplification, we get two factors for the given problem and solve them separately to get the required solution for the problem.

Complete step by step answer:
According to the problem, we have to find the value of x.
Thus, we will use the factoring method to get the solution.
The equation given to us is $5{{x}^{2}}=100$ ------- (1)
So, we will first divide 5 on both sides of the equation (1), we get
$\dfrac{5}{5}{{x}^{2}}=\dfrac{100}{5}$
On further simplification, we get
${{x}^{2}}=20$
Now, we will subtract 20 on both sides of the above equation, we get
${{x}^{2}}-20=20+20$
As we know, the same terms with opposite signs cancel out each other, therefore we get
${{x}^{2}}-20=0$
So, we will now apply the algebraic identity $(a+b)(a-b)={{a}^{2}}-{{b}^{2}}$ in the above equation, we get
$\left( x-\sqrt{20} \right)\left( x+\sqrt{20} \right)=0$
Thus, as we get two factors of the above equation, therefore we will solve them separately, that is
$\left( x-\sqrt{20} \right)=0$ --------- (2)
$\left( x+\sqrt{20} \right)=0$ ----------- (3)
Now, we will solve equation (2), which is
$\left( x-\sqrt{20} \right)=0$
Now, we will add $\sqrt{20}$ on both sides in the above equation, we get
$\left( x-\sqrt{20}+\sqrt{20} \right)=0+\sqrt{20}$
As we know, the same terms with opposite signs cancel out each other, therefore we get
$x=\sqrt{20}$
Now, we will solve equation (3), which is
$\left( x+\sqrt{20} \right)=0$
Now, we will subtract $\sqrt{20}$ on both sides in the above equation, we get
$\left( x+\sqrt{20}-\sqrt{20} \right)=0-\sqrt{20}$
As we know, the same terms with opposite signs cancel out each other, therefore we get
$x=-\sqrt{20}$

Therefore, for the equation $5{{x}^{2}}=100$ , the value of x is equal to $\sqrt{20},-\sqrt{20}$.

Note: While solving this problem, do mention all the steps properly to avoid confusion and mathematical errors. One of the alternative methods to solve this problem is first to subtract 100 on both sides of the equation and then take common 5 on both sides of the equation. After that, use the algebraic identity $(a+b)(a-b)={{a}^{2}}-{{b}^{2}}$ in the equation and solve further, to get the required result for the problem.