Question

How do you solve the equation $4{x^2} = 20x - 25$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. It becomes very easy to form the complete square if the given expression itself is the perfect square.

Complete step-by-step solution:
Take the given expression: $4{x^2} = 20x - 25$
Move all the terms from the right hand side of the equation to the left hand side of the equation. Remember when you move any term from one side to another, then the sign of the term also changes. Positive terms become negative and the negative term becomes positive.
$4{x^2} - 20x + 25 = 0$
The above equation can be re-written as: ${(2x)^2} - 2(2x)(5) + {(5)^2} = 0$
The above equation can be framed in the form of ${a^2} - 2ab + {b^2} = {(a - b)^2}$
${(2x - 5)^2} = 0$
Take the square root on both sides of the equation.
$\sqrt {{{(2x - 5)}^2}} = 0$
Square and square root cancel each other on the left hand side of the equation.
$ \Rightarrow 2x - 5 = 0$
Make “x” the subject and move constants on the right hand side of the equation.
$ \Rightarrow 2x = 5$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$ \Rightarrow x = \frac{5}{2}$
This is the required solution.

Note: Be careful about the sign convention and remember when you move any term from one side to another then the sign of the term also changes. Positive term becomes the negative and the negative term becomes positive.