Completing the Square

Completing the square rewrites any quadratic as a(x + p)² + q -- and unlike factorising, it always works, even when the roots aren't nice numbers.

$2x^2 + 4x - 1$

Factor 2 out of the x² and x terms:

$2\left(x^2 + 2x\right) - 1$

Half the coefficient of x (inside the bracket), then complete the square:

$2(x + 1)^2 - 3$

Turning point: $(-1, -3)$ -- a minimum because a is positive.

Solving $2x^2 + 4x - 1 = 0$

Real roots exist, but they're not nice numbers here (the discriminant, 24, isn't a perfect square) -- the quadratic formula would give exact surd answers, using the same turning point found above.