Question

How do you complete the square to solve $-{{x}^{2}}+6x+9=0$?

Answer 1

Hint: From the question we have been asked to find the solution to the equation $-{{x}^{2}}+6x+9=0$ using the completing square method. We can solve the given question by completing the square method by separating the variable terms in the equation from the constant terms and rearranging the equation $-{{x}^{2}}+6x+9=0$.

Complete step by step solution:
So, in the process of solving the question first we will remove the negative sign before ${{x}^{2}}$ by multiplying the equation with $-1$.
So, the equation will be reduced as follows.
$\Rightarrow -{{x}^{2}}+6x+9=0$
$\Rightarrow {{x}^{2}}-6x-9=0$
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 $18$ to the equation for the simplification. So, the equation will be reduced as follows.
$\Rightarrow {{x}^{2}}-6x-9+18-18=0$
$\Rightarrow {{x}^{2}}-6x+9-18=0$
Here now we will bring the integer $18$ to the right hand side of the equation. So, the equation will be reduced as follows.
$\Rightarrow {{x}^{2}}-6x+9=18$
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 reduced the equation as follows.
$\Rightarrow {{\left( x-3 \right)}^{2}}=18$
$\Rightarrow {{\left( x-3 \right)}^{2}}-18=0$
Therefore, in this way we complete the square $ {{\left( x-3 \right)}^{2}}-18=0$ to solve the question.
The solution for the equation will be as follows.
$\Rightarrow {{\left( x-3 \right)}^{2}}=18$
Here we will apply the square root on both sides.
$\Rightarrow \sqrt{{{\left( x-3 \right)}^{2}}}=\sqrt{18}$
$\Rightarrow x-3=\pm 3\sqrt{2}$
$\Rightarrow x=3\pm 3\sqrt{2}$
Therefore, the solution for the question using complete square method will be $\Rightarrow x=3\pm 3\sqrt{2}$ and in this way we complete the square $\Rightarrow {{\left( x-3 \right)}^{2}}-18=0$ to solve the question.

Note: Students must be careful in doing the calculations. Here we should be very careful and know that here we have two solutions as the square root to a constant will be $\pm 3\sqrt{2}$ and not just $3\sqrt{2}$. Therefore it has two answers using the completing square method.