Question

How do you solve \[{x^2} + 4x - 12 = 0\] by completing the square?

Answer 1

Hint: Here we will first add the square of half the value of \[x\] coefficient on both sides of the equation. Then by using the algebraic identity we will form the complete square on the left-hand side. Finally, we will solve for the value of \[x\] and get the required answer.

Complete step-by-step answer:
The equation given to us is:
\[{x^2} + 4x - 12 = 0\]…..\[\left( 1 \right)\]
To make the left hand side a perfect square we will add the square of half the value of \[x\] coefficient. Therefore,
Required value \[ = {\left( {\dfrac{4}{2}} \right)^2} = \dfrac{{16}}{4}\]
Dividing the terms, we get
\[ \Rightarrow \] Required value \[ = 4\]
Now we will add 4 on both sides of equation \[\left( 1 \right)\]. So, we get
\[{x^2} + 4x - 12 + 4 = 4\]
Adding 12 on both the sides, we get
\[\begin{array}{l} \Rightarrow {x^2} + 4x + 4 = 12 + 4\\ \Rightarrow {x^2} + 4x + 4 = 16\end{array}\]
Next, using algebraic identity \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\] on left side, we get
\[\begin{array}{l} \Rightarrow {x^2} + 2 \times 2 \times x + {2^2} = 16\\ \Rightarrow {\left( {x + 2} \right)^2} = 16\end{array}\]
Taking the square root both sides we get,
\[\begin{array}{l} \Rightarrow x + 2 = \sqrt {16} \\ \Rightarrow x + 2 = \pm 4\end{array}\]
Subtracting 2 from both the sides, we get
\[ \Rightarrow x = \pm 4 - 2\]
We get our two values as,
\[ \Rightarrow x = 4 - 2\] and \[x = - 4 - 2\]
Adding and subtracting the terms, we get
\[ \Rightarrow x = 2\] and \[x = - 6\]

Therefore, the solution of the given equation is \[x = 2\] and\[x = - 6\].

Note:
A quadratic equation is an equation having the variable with the highest power as two. The method that is always applicable to find the factors of an equation is known as the completing square method and it is used when the other two methods i.e. factoring and square root method can’t be applied. As the highest power of the variable is two we get two factors for the given equation. In the complete square method, we check what constant term should be added to the existing two terms of a square so that it can be made a perfect square.