Question

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

Answer 1

Hint: Completing squares is the process of making an expression turn into a perfect square. This one of the methods of solving for $x$. We can solve for $x$ either by splitting the mid term or by using the quadratic equation formula which is $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Either way we get the answer we want.

Complete step by step answer:
So now let’s try to complete the equation ${{x}^{2}}+4x+2=0$.
We know that this equation sort of resembles ${{\left( x+2 \right)}^{2}}$ .
${{\left( x+2 \right)}^{2}}$ expands to ${{x}^{2}}+4x+4$ .
So we can try to convert the given equation in the question to ${{\left( x+2 \right)}^{2}}$ as they look almost the same.
So now let’s add and subtract a 2 on the left hand side of the question. Upon doing so we get the following :
$\begin{align}
  & {{x}^{2}}+4x+2=0 \\
 & {{x}^{2}}+4x+2+2-2=0 \\
\end{align}$
We can do this as this does not change the value of our equation in any way whatsoever.
Now , let’s send the -2 onto the right hand side of the equation from the left hand side while keeping all the others on the right itself.
Upon doing so, we get the following :
$\begin{align}
  & {{x}^{2}}+4x+2+2-2=0 \\
 & {{x}^{2}}+4x+4=2 \\
 & \\
\end{align}$
So now we can shrink ${{x}^{2}}+4x+4$ this to ${{\left( x+2 \right)}^{2}}$ . Upon doing so , we get the following :
$\Rightarrow {{\left( x+2 \right)}^{2}}=2$ .
Let’s take a square root on both sides. Upon doing so, we get the following :
$\Rightarrow \left( x+2 \right)=\pm \sqrt{2}$
Now let’s send 2 from the left hand side of the equation to the right hand side. Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow \left( x+2 \right)=\pm \sqrt{2} \\
 & \Rightarrow x=\pm \sqrt{2}-2 \\
\end{align}$
So now we got two values of $x$ and they are the following :
$\begin{align}
  & \Rightarrow x=\sqrt{2}-2 \\
 & \Rightarrow x=-\sqrt{2}-2 \\
\end{align}$

$\therefore $ The values of $x$ that we got upon solving the equation ${{x}^{2}}+4x+2=0$ using completing squares are $\sqrt{2}-2$ and $-\sqrt{2}-2$.

Note: We have to be very careful while completing the squares. We should know what is the closest square to our given question. The complete square which almost resembles our question. For this , a lot of practice is required. And after we figure out this , then we have to add and subtract the necessary constants. And don’t forget that for a quadratic equation, we always have two values of $x$ .