Question

How do you solve the equation by completing the square ${x^2} - 14x + 33 = 0$?

Answer 1

Hint: According to the question, we have to solve the equation by completing the square ${x^2} - 14x + 33 = 0$. So, first of all we have to take the constant term of the given equation as ${x^2} - 14x + 33 = 0$ to the right hand side of that equation.
Now, we have to make the left hand side of the equation a perfect square by adding the third term to the both sides of the expression obtained.

Complete step-by-step solution:
Step 1: First of all we have to take the constant term of the given equation as ${x^2} - 14x + 33 = 0$ to the right hand side of that equation as mentioned below.
$ \Rightarrow {x^2} - 14x = - 33$
Step 2: So, first of all we have to find the third term by using the formula (A) as mentioned in the solution hint.
$ \Rightarrow $Third term $ = {\left( {\dfrac{1}{2} \times 14} \right)^2}$
$ \Rightarrow 49$
Step 3: Now, we have to make the left hand side of the expression obtained in the solution step 1 is a perfect square by adding the third term as obtained in the solution step 2 to the both sides of the expression obtain in the solution step 1.
$ \Rightarrow {x^2} - 14x + 49 = - 33 + 49$
Now, we have to solve the expression obtain just above,
$ \Rightarrow {\left( {x - 7} \right)^2} = 16$
Now, we have to take square roots on both sides of the expression obtain just above,
$
   \Rightarrow \sqrt {{{\left( {x - 7} \right)}^2}} = \sqrt {16} \\
   \Rightarrow \left( {x - 7} \right) = \pm 4
 $
Step 4: Now, we have to find the values of $x$ by equating the expression obtained in the solution step 3 to the $ + 4$and $ - 4$.
$ \Rightarrow \left( {x - 7} \right) = + 4$
$
   \Rightarrow x = 4 + 7 \\
   \Rightarrow x = 11
 $
And,
$ \Rightarrow \left( {x - 7} \right) = - 4$
$
   \Rightarrow x = - 4 + 7 \\
   \Rightarrow x = 3
 $

Hence, the solution of the equation by completing the square ${x^2} - 14x + 33 = 0$ is $x = 11$ and $x = 3$.

Note: It is necessary to make the given equation as ${x^2} - 14x + 33 = 0$ is a perfect square to solve that equation.
It is necessary to find a third term that makes the given equation a perfect square.