Question

How do you solve the equation ${x^2} - 6x + 9 = 8$ by completing the square?

Answer 1

Hint: Completing the square is the method which represents the quadratic equation as the combination of the quadrilateral used to form the square and it is the basis of the method discovers the special value which when added to both the sides of the quadratic which creates the perfect square trinomial. Here we will take the given expression and check for the perfect square or the value to be added.

Complete step-by-step solution:
Take the given expression: ${x^2} - 6x + 9 = 8$
By observing the above equation which states that the left hand side of the given equation is the perfect square comparing with standard form ${x^2} - 2ax + {a^2} = {(x - a)^2}$
${(x - 3)^2} = 8$
Take the square root on both the sides of the equation.
$\sqrt {{{(x - 3)}^2}} = \sqrt 8 $
The above equation can be re=written as –
$\sqrt {{{(x - 3)}^2}} = \sqrt {4 \times 2} $
Square and square root cancel each other on the left hand side of the equation. Always remember that the square root of any positive term can be positive or the negative.
$ \Rightarrow x - 3 = \pm 2\sqrt 2 $
Make the required term “x” as the subject and move the other term on the opposite side. When you move any term from one side to another, then the sign of the term also changes. Negative terms become positive and vice-versa.
$ \Rightarrow x = 3 \pm 2\sqrt 2 $
This is the required solution.

Note: Be careful about the sign convention, when you move any term from one side to the other the sign of the term also changes. Positive terms become negative and the negative term becomes positive. Also, remember the square of positive and the negative term gives a result always as positive and square root of the positive term can give negative or positive terms.