How do you solve x2-x-c by completing the square?

Answer 1

To complete the square, divide the coefficient of the x term by 2, put this in the square and subtract it squared (at the end) to keep the value the same.

For x2 - x - c:

• the coefficient of the x is -1
• half of this is -1/2
• so the square is (x - 1/2)2
• and the -1/2 squared is (-1/2)2 = 1/4

and the equation becomes:

x2 - x - c = (x - 1/2)2 -1/4 - c

And if the equation equals 0, then the equation can be solved:

x2 - x - c = 0

⇒ (x - 1/2)2 -1/4 - c = 0

⇒ (x - 1/2)2 = 1/4 + c

⇒ x - 1/2 = ±√(1/4 + c)

⇒ x = 1/2 ± √(1/4 + c)

So:

• if c < -1/4, there are no real solutions
• if c = -1/4, there is one (repeated)) real solution
• if c > -1/4, there are two real solutions