Question

How do you solve the equation ${{x}^{2}}-16x=0$ by a complete square method?

Answer 1

Hint: Now to solve the equation by complete square method we will first make sure if the coefficient of ${{x}^{2}}$ is 1. Then to the equation we add and subtract the term ${{\left( \dfrac{b}{2a} \right)}^{2}}$ Now after simplifying the equation by using formula ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ we will equation in form ${{\left( x-a' \right)}^{2}}=b'$ . Hence taking the square root we can easily find the solution of the given equation.

Complete step-by-step answer: Now consider the given equation ${{x}^{2}}-16x=0$ .
We will solve this equation by complete square method.
Now first we want the coefficient of ${{x}^{2}}$ to be 1. Since the coefficient is already 1 we can proceed further. Now we want to create a perfect square in the equation. To do so we will have to add and subtract terms such that we get a perfect square. First let us compare the equation with the general quadratic equation $a{{x}^{2}}+bx+c=0$ Hence we get a = 1, b = -16 and c = 0.
Now we will add and subtract the term ${{\left( \dfrac{b}{2a} \right)}^{2}}$ to the equation, Hence we get,
$\begin{align}
  & \Rightarrow {{x}^{2}}-16x+{{\left( \dfrac{16}{2\left( 1 \right)} \right)}^{2}}-{{\left( \dfrac{16}{2\left( 1 \right)} \right)}^{2}}=0 \\
 & \Rightarrow {{x}^{2}}-16x+64-64=0 \\
\end{align}$
Hence we have $\Rightarrow {{x}^{2}}-2\times 8x+{{8}^{2}}=64$
Now we know that ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ hence using this formula in the above equation we get,
$\Rightarrow {{\left( x-8 \right)}^{2}}=64$
Now taking square root on both sides of the equation we get, $x-8=\pm 8$
Hence we have x = 0 or x = 16.
Hence the solution of the given equation is x = 0 and x = 16.

Note: Note that while taking square root in the equation we get two cases one for positive value and one for negative value. Hence we need to consider both cases to solve the equation. Also for a quadratic equation if c = 0 we can easily solve it by taking x common. Hence we can easily solve the equation by writing the equation as $x\left( x-16 \right)=0$.