Question

How do you solve \[{{x}^{2}}+6x+2=0\]?

Answer 1

Hint: In this problem, we have to solve and find the value of x. We have to check whether the given equation is a perfect square, if not we have to convert it to a perfect square to find the value of x. We can also use the quadratic formula method, by comparing the general quadratic equation to the given equation to find the variables for the formula to be used.

Complete step by step answer:
We know that the given quadratic equation to be solved to find the value of x is,
 \[{{x}^{2}}+6x+2=0\]
We can now subtract -2 on both the left-hand side and the right-hand side, we get
\[\begin{align}
  & \Rightarrow {{x}^{2}}+6x+2-2=0-2 \\
 & \Rightarrow {{x}^{2}}+6x=-2 \\
\end{align}\]
Now we can add the number 9 on both the left-hand side and the right-hand side, we get
\[\begin{align}
  & \Rightarrow {{x}^{2}}+6x+9=-2+9 \\
 & \Rightarrow {{x}^{2}}+6x+9=7 \\
\end{align}\]
We know that the algebraic formula is,
\[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
Now we can apply this formula in the above step, we get
\[\Rightarrow {{\left( x+3 \right)}^{2}}=7\]
Now we can take square root on both the left-hand side and the right-hand side, we get
\[\Rightarrow x+3=\pm \sqrt{7}\]
Now we can subtract the number 3 on both the sides, we get
\[\Rightarrow x=-3\pm \sqrt{7}\]

Therefore, the value of \[x=-3+\sqrt{7},-3-\sqrt{7}\]

Note: Students make mistakes while adding/subtracting the values in both the sides of the equation in order to simplify the equation to get the value of x. We should also know some algebraic formulas and identities to solve these types of problems. We can also solve this problem using quadratic formulas.