Question

How do you complete the square to solve ${{x}^{2}}+6x+34=0$?

Answer 1

Hint: From the question we have been asked to find the solution to the equation ${{x}^{2}}+6x+34=0$ using the completing square method. The complete square method means we will get the constant right side of the equation and the variables in the form of square will be present at the left hand side. The strict rule is that to apply a complete square method the coefficient of ${{x}^{2}}$should be one and positive then only we can apply this complete square method.

Complete step by step solution:
From the question given we have to solve the ${{x}^{2}}+6x+34=0$ by complete square method.
$\Rightarrow {{x}^{2}}+6x+34=0$
Here we must make sure that the coefficient of ${{x}^{2}}$ should be 1 and nothing other than that, from the above equation we can clearly see that its coefficient is 1. So, we will further continue our solution as follows.
Here we will add and subtract the integer $9$ to the equation for the simplification. So, the equation will be reduced as follows.
$\Rightarrow {{x}^{2}}+6x+34=0$
$\Rightarrow {{x}^{2}}+6x+34+9-9=0$
Here now we will bring the integer $25$ to the right-hand side of the equation. So, the equation will be reduced as follows.
$\Rightarrow {{x}^{2}}+6x+9=-25$
Now, here we can clearly see that the expression on the left-hand side of the equation is a complete square. So, after rewriting the equation we will get the equation reduced as follows.
$\Rightarrow {{\left( x+3 \right)}^{2}}=-25$
Therefore, in this way we complete the square $\Rightarrow {{\left( x+3 \right)}^{2}}=-25$ to solve the question.
The solution for the equation will be as follows.
$\Rightarrow {{\left( x+3 \right)}^{2}}=-25$
Here we will shift the square of the left-hand side to the right-hand side then it will be square root to the right-hand side equation.
$\Rightarrow \left( x+3 \right)=\pm \sqrt{-25}$
We know that the square root of $-1$ is i.
$\Rightarrow \left( x+3 \right)=\pm 5i$
$\Rightarrow x=\pm 5i-3$
Therefore, the solution for the question using the complete square method will be $\Rightarrow x=\pm 5i-3$ and in this way we complete the square ${{x}^{2}}+6x+34=0$ to solve the question.

Note: Students should know this method because it is one of the most important methods. Students should check the coefficient of ${{x}^{2}}$ should be 1 and nothing other than that. Students should consider that minus sign in the square root if they neglected the square root then the whole answer will be wrong. So, students must be careful in this aspect.