Question

How do you solve the quadratic equation ${{x}^{2}}-5x=0$ by completing the square?

Answer 1

Hint: We start solving the problem by adding both sides of the given quadratic equation with $\dfrac{25}{4}$. We then make the necessary calculations after addition and then make use of the fact that ${{a}^{2}}-2ab+{{b}^{2}}={{\left( a-b \right)}^{2}}$ to proceed through the problem. We then apply the square root on both sides of the obtained result and then make the necessary calculations to get the required solution(s) of the given quadratic equation.

Complete step-by-step answer:
According to the problem, we are asked to solve the quadratic equation ${{x}^{2}}-5x=0$ by completing the square.
We have given the quadratic equation ${{x}^{2}}-5x=0$ ---(1).
Let us add both sides of the equation (1) with $\dfrac{25}{4}$.
$\Rightarrow {{x}^{2}}-5x+\dfrac{25}{4}=0+\dfrac{25}{4}$.
$\Rightarrow {{x}^{2}}-2\left( \dfrac{5}{2} \right)x+{{\left( \dfrac{5}{2} \right)}^{2}}=\dfrac{25}{4}$ ---(2).
We can see that the equation ${{x}^{2}}-2\left( \dfrac{5}{2} \right)x+{{\left( \dfrac{5}{2} \right)}^{2}}$ resembles ${{a}^{2}}-2ab+{{b}^{2}}$. We know that ${{a}^{2}}-2ab+{{b}^{2}}={{\left( a-b \right)}^{2}}$. Let us use this result in equation (2).
\[\Rightarrow {{\left( x-\dfrac{5}{2} \right)}^{2}}={{\left( \dfrac{5}{2} \right)}^{2}}\].
\[\Rightarrow x-\dfrac{5}{2}=\pm \dfrac{5}{2}\].
Let us assume \[x-\dfrac{5}{2}=\dfrac{5}{2}\].
\[\Rightarrow x=\dfrac{5}{2}+\dfrac{5}{2}\].
\[\Rightarrow x=\dfrac{10}{2}\].
\[\Rightarrow x=5\].
Let us assume \[x-\dfrac{5}{2}=-\dfrac{5}{2}\].
\[\Rightarrow x=-\dfrac{5}{2}+\dfrac{5}{2}\].
\[\Rightarrow x=0\].
So, we have found the solution(s) of the given quadratic equation ${{x}^{2}}-5x=0$ as $x=5$, $x=0$.
$\therefore $ The solution(s) of the given quadratic equation ${{x}^{2}}-5x=0$ are $x=5$, $x=0$.

Note: We should perform each step carefully in order to avoid confusion and calculation mistakes while solving this problem. We can also solve the given problem by factoring the given quadratic equation and then equating the obtained factors to 0 to get the solution(s) of the given quadratic equation. Whenever we get this type of problems, we convert the L.H.S (Left Hand Side) of the given quadratic equation to ${{\left( a-b \right)}^{2}}$ or ${{\left( a+b \right)}^{2}}$ to get the required answer. Similarly, we can expect problems to find the solution of the given quadratic equation: $9{{x}^{2}}-9x+18=0$.