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.