Question

Given the equation $\left( ax+b \right)\left( ax-b \right)=0$, how do you solve for $x$?

Answer 1

Hint: In this problem we have given the equation and asked to solve the equation for $x$. We can observe that the given equation is a quadratic equation which is factorized. To solve the quadratic equation, we either factorize it or use the formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$. In this problem we have directly factored quadratic equations, so there is no need to factorize it. Now to solve the equation, we are going to equate each factor to zero individually. Now we will get two equations separately. We will apply some basic arithmetic operations to solve both the equations. After solving the both the equations we can write the both the values as the solution of the given equation.

Complete step by step answer:
Given equation $\left( ax+b \right)\left( ax-b \right)=0$.
We can observe that the above equation is factored quadratic equation. To solve the quadratic equation, we are going to equate each factor to zero individually, then we will get
$ax+b=0$ or $ax-b=0$.
Considering the equation $ax+b=0$. In the above equation we have $b$ in addition, so subtracting $b$ from both sides of the equation, then we will get
$\begin{align}
  & \Rightarrow ax+b-b=-b \\
 & \Rightarrow ax=-b \\
\end{align}$
In the above equation we have $a$ in multiplication, so dividing the above equation with $a$ on both sides of the equation, then we will have
$\begin{align}
  & \Rightarrow \dfrac{ax}{a}=-\dfrac{b}{a} \\
 & \Rightarrow x=-\dfrac{b}{a} \\
\end{align}$
So, the solution for the equation $ax+b=0$ is $x=-\dfrac{b}{a}$.
Considering the equation $ax-b=0$. In this equation we have $b$ in subtraction, so adding $b$ on both sides of the above equation, then we will get
$\begin{align}
  & \Rightarrow ax-b+b=b \\
 & \Rightarrow ax=b \\
\end{align}$
In the above equation we have $a$ in multiplication, so dividing the above equation with $a$ on both sides of the equation, then we will have
$\begin{align}
  & \Rightarrow \dfrac{ax}{a}=\dfrac{b}{a} \\
 & \Rightarrow x=\dfrac{b}{a} \\
\end{align}$
So, the solution for the equation $ax-b=0$ is $x=\dfrac{b}{a}$.

Hence from the solutions of $ax+b=0$, $ax-b=0$. The solution of the given equation will be $x=\pm \dfrac{b}{a}$.

Note: We can also use another method to solve the above equation. We have the given equation
$\left( ax+b \right)\left( ax-b \right)=0$
We have the formula $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$. Applying this formula in the above equation, then we will get
${{\left( ax \right)}^{2}}-{{b}^{2}}=0$
Simplifying the above equation, then we will have
$\begin{align}
  & \Rightarrow {{a}^{2}}{{x}^{2}}-{{b}^{2}}=0 \\
 & \Rightarrow {{a}^{2}}{{x}^{2}}={{b}^{2}} \\
 & \Rightarrow {{x}^{2}}=\dfrac{{{b}^{2}}}{{{a}^{2}}} \\
 & \Rightarrow x=\sqrt{\dfrac{{{b}^{2}}}{{{a}^{2}}}} \\
 & \Rightarrow x=\pm \dfrac{b}{a} \\
\end{align}$
From both the methods we got the same result.