completing the square

Question

w^2-w-20=0

Answer 1

w² - w - 20 = 0

 

When we complete the square, we want the left side of the equation to be a polynomial that is a product of a square.  We look at the first and middle terms to determine what to do with the equation. 

 

-(1/2)w - (1/2)w = -w

 

Therefore, the square will be (w - (1/2))².  The last term must be +1/4.  We need to find a value that when -20 is added to it, the result is +1/4.

 

-20 + x = 1/4

         x = (1/4) + 20

         x = (1 + 80) / 4

         x = 81 / 4

 

 

Add 81/4 on both sides of equation.

 

w² - w - 20 + (81/4) = 0 + (81 / 4)

 

w² - w + 1/4 = 81/4

 

(w - (1/2))² = 81/4

 

 

Square-root both sides of equation.

 

w - 1/2 = ±√(81/4)

 

w = (1/2) ± √(81/4)

 

w = (1/2) ± (9/2)

 

w = (1/2) + (9/2)               and          w = (1/2) - (9/2)

 

w = 10/2                            and          w = -8/2

 

w = 5                               and           w = -4