Completing the Square

# Completing the Square

GCSE(H),

Quadratic equations can be solved by completing the square. Get the equation in the form ax^2^+bx+c, and using an example of 3x^2+6x-4:

Factorise the x^2 and x terms with the a term: 3(x^2 + 2x) - 4

Divide the new b term by 2 to create a squared term: 3((x+1)^2 - 1 ) - 4

Multiply out the factor: 3(x+1)^2 - 3 - 4 = 3(x+1)^2 - 7 = 0

Solve this equation:

3(x+1)^2 = 7

(x+1)^2 = frac(7)(3)

x+1 = +-sqrt(frac(7)(3)

x = +-sqrt(frac(7)(3)) - 1

## Examples

1. By completing the square, solve, giving the answer in surd form, x^2 +2x - 2 = 0.

Answer: x=+-sqrt(3) - 1

x^2 + 2x - 2 = 0

((x + 1)^2 - 1) - 2 = 0

(x+1)^2 = 3

(x + 1) = sqrt(3)

x=+-sqrt(3) - 1

Check the answer by substituting back

2. Explain, by completing the square, why x^2 - 2x + 4 = 0 has no solution.

Answer: The solution involves the square root of a negative number

x^2-2x+4=0

((x-1)^2 -1) + 4 = 0

(x -1)^2 + 3 = 0

Attempting to solve the equation: (x-1)^2 = -3

(x-1) = sqrt(-3), which is an attempt to find the square root of a negative number.