Question

How do you solve $ 4{{x}^{2}}+40x+280=0 $ by completing the square?

Answer 1

Hint:
We will divide the equation by the coefficient of $ {{x}^{2}} $ . Then we will shift the constant term to the other side of the equation. Then we will complete the square on the left hand side and balance it by adding the same term on the right and side. After that we will take the square root on both the sides and find the value of the variable.

Complete step by step answer:
The given quadratic equation is $ 4{{x}^{2}}+40x+280=0 $ . We have to solve this equation and find the value of $ x $ by using the completing square method. The first step in this method is to divide the equation by the coefficient of $ {{x}^{2}} $ , which is 4 for the given quadratic equation. Dividing the given quadratic equation by 4, we get the following equation,
 $ \begin{align}
  & \dfrac{4{{x}^{2}}}{4}+\dfrac{40x}{4}+\dfrac{280}{4}=0 \\
 & \therefore {{x}^{2}}+10x+70=0 \\
\end{align} $
Next, we will shift the constant term to the other side of the equation. So, we have
 $ {{x}^{2}}+10x=-70 $
Now, we have to complete the square on the left hand side. This means that the left hand side should resemble the expression $ {{x}^{2}}+2xy+{{y}^{2}} $ . So, we have to find the $ y $ for our equation.
We can see that $ 10x=2x\times 5 $ . So, we have $ y=5 $ . Therefore, $ {{y}^{2}}=25 $ . We can complete the square on the right hand side by adding the term 25. We will balance the equation by adding 25 to the right hand side. So, we have
 $ {{x}^{2}}+10x+25=-70+25 $
Since the left hand side is of the form $ {{x}^{2}}+2xy+{{y}^{2}} $ and we know that $ {{x}^{2}}+2xy+{{y}^{2}}={{\left( x+y \right)}^{2}} $ ,
we can write the above equation as
 $ {{\left( x+5 \right)}^{2}}=-45 $
Taking square root on both sides, we get
 $ \begin{align}
  & x+5=\sqrt{-45} \\
 & \Rightarrow x+5=\pm 3\sqrt{5}i \\
 & \therefore x=-5\pm 3\sqrt{5}i \\
\end{align} $
These are the roots of the given equation.

Note:
The completing square method is one of the methods to solve a quadratic equation. The other methods are factorization method and using quadratic formula. We can use one of these two methods to verify the answer obtained by using the completing square method. It is useful to do the calculations explicitly so that we can avoid making minor errors.