# Complete the Square for a^2-a+6

Enter Quadratic equation/inequality below

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Complete the square for the quadratic a2-a+6

With the standard form of ax2 + bx + c, we have our a, b, and c values:
a = 1, b = 1, c = 6

Solve the quadratic a2 + a + 6 by completing the square

We first subtract 6 to isolate our variable on the left side
a2 + a + 6 - 6 = 0 - 6
a2a = -6

Complete the square by adding an amount to both sides:
a2 + 1a + ? = -6 + ?
The amount to add to each side is (½ * middle term coefficient)2
 Amount to add = (1 x 1)2 (2 x 1)2

 Amount to add = (1)2 (2)2

 Amount to add = 1 4

Amount to add = 1/4

Rewrite our perfect square equation:
a2 + 1 + (1/2)2 = -6 + (1/2)2
(a + 1/2)2 = -6 + 1/4

Simplify Right Side of the Equation:
Using our LCM Calculator, the Least Common Multiple of our denominators 1 and 4 = 4

We multiply -6 by 4 ÷ 1 = 4 and 1 by 4 ÷ 4 = 1
 Simplified Fraction = -6 x 4 + 1 x 1 4

 Simplified Fraction = -24 + 1 4

 Simplified Fraction = -23 4

We set our left side of the complete the square equation = u
u2 = (a + 1/2)2

u has two solutions:
u = +√-23/4
u = -√-23/4

Replacing u with the equation it represents, we get:
a + 1/2 = +NAN
a + 1/2 = -NAN

Subtract 1/2 from the both sides of Solution Equation 1
a + 1/2 - 1/2 = +√/1 - 1/2
Simplify right side of the equation
Using our LCM Calculator, the Least Common Multiple of our denominators 1 and 2 = 2

We multiply √ by 2 ÷ 1 = 2 and -1 by 2 ÷ 2 = 1
 Simplified Fraction = √ x 2 - 1 x 1 2

 Simplified Fraction = 0 - 1 2

 Simplified Fraction = -1 2

Answer 1 = -1/2

Subtract 1/2 from the both sides of Solution Equation 2
a + 1/2 - 1/2 = -√/1 - 1/2
Simplify right side of the equation
Using our LCM Calculator, the Least Common Multiple of our denominators 1 and 2 = 2

We multiply 0 by 2 ÷ 1 = 2 and -1 by 2 ÷ 2 = 1
 Simplified Fraction = 0 x 2 - 1 x 1 2

 Simplified Fraction = 0 - 1 2

 Simplified Fraction = -1 2

Answer 2 = -1/2