Why complete the square when the quadratic formula always works?

Because the formula gives you the roots and nothing else. Completing the square gives you the turning point, the line of symmetry, the range, and whether there are real roots at all — and questions ask for those far more often than they ask you to solve.

What the form tells you

Written as a(x − h)² + k, every feature of the parabola is visible:

• Turning point at (h, k)
• Line of symmetry is x = h
• Minimum value is k if a > 0; maximum is k if a < 0
• Range is y ≥ k or y ≤ k accordingly

Note the sign: x² − 6x + 11 becomes (x − 3)² + 2, and the turning point is at x = +3, not −3. The bracket says "x minus three", so the shift is three to the right. This sign flip is the single most common error in the topic.

A worked example

Take x² − 6x + 11.

Halve the coefficient of x to get −3, square it to get 9:

x^2 - 6x + 11
= (x - 3)^2 - 9 + 11
= (x - 3)^2 + 2

Minimum value 2, at x = 3. Since 2 > 0 the curve never reaches the x-axis, so there are no real roots — and you have established that without a discriminant.

When the coefficient of x² is not 1, factor it out of the first two terms first:

2x^2 + 8x + 5
= 2(x^2 + 4x) + 5
= 2[(x + 2)^2 - 4] + 5
= 2(x + 2)^2 - 8 + 5
= 2(x + 2)^2 - 3

The −4 inside the bracket gets multiplied by the 2 on the way out. Forgetting that is the second most common error.

Which method for which question

Factorising — fastest when the quadratic factorises cleanly. Always try it first; it takes a few seconds to find out.

Completing the square — when the question asks for the turning point, the maximum or minimum, the range, or to sketch. Also when it says "leave your answer in surd form", because the method produces exact values naturally.

The formula — when it does not factorise and you only need the roots. Reliable, slower, and tells you nothing about the shape.

The discriminant alone — when the question asks only how many roots there are, or for a condition on an unknown coefficient. No need to solve anything.

Where it earns the most

Optimisation. A question giving you an expression for area or profit and asking for the maximum is asking you to complete the square — the turning point is the answer, and no calculus is needed.

Also inequalities. Writing x² − 6x + 11 > 0 as (x − 3)² + 2 > 0 makes it immediately obvious the statement is true for all x, because a square is never negative and 2 is positive. Proving that from the original form is considerably more work.