Question

How do you solve completing the square $2{{x}^{2}}+4x+12=0$.

Answer 1

Hint: By solving completing the square first divide all terms by coefficient of ${{x}^{2}},$ then move the number tern $(c/a)$ to the right of the equation. Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

Complete Step by Step solution:
The given equation:
$2{{x}^{2}}+4x+1=0...(i)$
The equation is in the form of quadratic equation:
\[\Rightarrow \]$a{{x}^{2}}+bx+c=0$
Hence,
From equation $(i)$
\[\Rightarrow \]$2{{x}^{2}}+4x+12=0$
\[\Rightarrow \]$2{{x}^{2}}+4x=-12$
Divide both side by $2$
\[\Rightarrow \]$\dfrac{2}{2{{x}^{2}}}+\dfrac{4x}{2}=\dfrac{-12}{2}$
\[\Rightarrow \]${{x}^{2}}+\dfrac{4}{2}x=\dfrac{-12}{2}$
\[\Rightarrow \]${{x}^{2}}+2x=-5$
Adding and subtracting $1$ in L.H.S.
\[\Rightarrow \]${{x}^{2}}+2x+1-1=-6$
\[\Rightarrow \]${{\left( x+1 \right)}^{2}}=-6+1$
Here using formula
\[\Rightarrow \]${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$
\[\Rightarrow \]${{\left( x+1 \right)}^{2}}=-5$
Taking square root both side
\[\Rightarrow \]$\sqrt{{{\left( x+2 \right)}^{2}}}=\pm \sqrt{-5}$
\[\Rightarrow \]$\left( x+2 \right)=\pm \sqrt{-5}$
Since,
We have a negative square root. Did you see complex numbers get $2$ if not stop here and say the real solution.
If yes
\[\Rightarrow \]$x=-1\pm i\sqrt{5}$

Additional Information:
When you have a polynomial such as
\[\Rightarrow \]${{x}^{2}}+4x+20$
It is sometimes desirable to express it in the form of
\[\Rightarrow \]${{a}^{2}}+{{b}^{2}}$
To do this, we can artificially introduce a constant which allows us to factor a perfect square out of the expression like so,
\[\Rightarrow \]${{x}^{2}}+4x+20$
$={{x}^{2}}+4x+4-4+20$
Notice that by simultaneously adding and subtracting $4,$ we have not changed the value of the expression.
Now, we can do this:
$=\left( {{x}^{2}}+4x+4 \right)+\left( 20-4 \right)$
$={{\left( x+2 \right)}^{2}}+16$
$={{\left( x+2 \right)}^{2}}+{{4}^{2}}$
We have completed the square.
And completing the square useful to simplify quadratic expression so that they become soluble with square roots completing the square is an example of four transformations the use of a substitution (albeit implicitly) in reducing a polynomial equation to a simple formula.

Note:
Always make sure to first divide each term by the coefficient of\[{{x}^{2}}\]. Always make sure that the coefficient square term must be\[1\]. If not then divide it into a suitable number for obtaining\[1\]. Always do correct algebraic operations. Remember some basic formula \[{{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]for solving or simplifying the questions. These are some of the important points we need to remember for solving this type of question.