Completing the Square (a=1)

Statement

Completing the square is a method for rewriting a quadratic expression in a form that reveals its minimum/maximum value and turning points. For a quadratic with leading coefficient 1, the identity is:

\[ x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b^2}{4} - c\right) \]

This expresses the quadratic as a perfect square plus or minus a constant, making it easier to solve, graph, or analyse.

Why it’s true

• A perfect square trinomial has the form \((x+p)^2 = x^2 + 2px + p^2\).
• Given \(x^2 + bx\), we can match \(b = 2p\), so \(p = b/2\).
• Adding and subtracting \((b/2)^2\) completes the square: \(x^2+bx = (x+b/2)^2 - (b/2)^2\).
• Then we add the constant \(c\), giving the full identity.

Recipe (how to use it)

1. Take the coefficient of \(x\) (which is \(b\)).
2. Halve it (\(b/2\)) and square it (\((b/2)^2\)).
3. Add and subtract this value inside the expression.
4. Write the first three terms as a square, simplify the rest.

Spotting it

Whenever you are asked to solve a quadratic by completing the square, find the vertex of a parabola, or rewrite in the form \((x+p)^2+q\), this technique applies.

Common pairings

• Solving quadratic equations (leading to the quadratic formula).
• Finding maximum/minimum values in optimisation problems.
• Sketching quadratic graphs by locating the vertex.

Mini examples

1. \(x^2+6x+5=(x+3)^2-9+5=(x+3)^2-4\).
2. \(x^2-4x+7=(x-2)^2-4+7=(x-2)^2+3\).

Pitfalls

• Forgetting to subtract the square: Always balance the equation by subtracting what you added.
• Sign errors: Be careful with negative \(b\); halving and squaring always makes the last term positive.
• Wrong halving: Remember it’s half of \(b\), not half of \(c\).

Exam strategy

• Write down the process step by step to avoid mistakes.
• Check your final answer by expanding back to the original quadratic.
• Estimate: the constant term in the completed form should be close in size to the original \(c\).

Summary

Completing the square is a method to rewrite a quadratic \(x^2+bx+c\) into the form \((x+p)^2+q\), revealing the vertex of the parabola and allowing solutions to be found more easily. It is central to graph sketching and solving equations when factorisation is not straightforward.