Question

How do you solve $ {y^2} - 5y - 3 = 0 $ by completing the square?

Answer 1

Hint: To solve the given equation by completing the square, we should go through the difference of squares identity and then solve for the equation. And then we will discuss the steps about the method of finding roots by completing the squares.

Complete step by step solution:
Given equation: $ {y^2} - 5y - 3 = 0 $
As we can see that, the given equation is a quadratic equation.
We can use the difference of squares identity, which can be written as:
 $ {a^2} - {b^2} = (a - b)(a + b) $
Pre-multiply by 4 to reduce the amount of working in fractions:
 $
  0 = 4({y^2} - 5y - 3) \\
 = 4{y^2} - 20y - 12 \\
= 4{y^2} - 20y + 25 -12 - 25 \\
 = {(2y - 5)^2} - 25 - 12 \\
 = {(2y - 5)^2} - {(\sqrt {37} )^2} \\
 = ((2y - 5) - \sqrt {37} )((2y - 5) + \sqrt {37} ) \\
 = (2y - 5 - \sqrt {37} )(2y - 5 + \sqrt {37} ) \;
 $
with $ a = 2y - 5 $ and $ b = \sqrt {37} $ .
Hence,
 $ y = \dfrac{5}{2} \pm \dfrac{{\sqrt {37} }}{2} $
So, the correct answer is “ $ y = \dfrac{5}{2} \pm \dfrac{{\sqrt {37} }}{2} $ ”.

Note: Steps for completing the squares:
Step-1: Write the equation in the form, such that c is on the right side.
Step-2: If $ a $ is not equal to 1, then divide the complete equation by a, such that coefficient of $ {x^2} $ is 1.
Step-3: Now add the square of half of the coefficient of term-x, (b/2a) 2, on both the sides.
Step-4: Factorize the left side of the equation as the square of the binomial term.
Step-5: Take the square root on both the sides
Step-6: Solve for variable x and find the roots.