Cambridge O Level Additional Mathematics · Syllabus 4037 · Quadratic Functions
Completing the Square
What is Completing the Square?
An algebraic rearrangement that rewrites a quadratic ax^2 + bx + c in the equivalent form a(x - h)^2 + k. Because a squared bracket is never negative, the constant k is the minimum value of the function when a is positive and the maximum value when a is negative, attained at x = h; the point (h, k) is therefore the turning point of the parabola. The rearrangement changes only how the expression is written, never its value at any x.
This definition is part of the Quadratic Functions chapter in Cambridge O Level Additional Mathematics.
Questions students ask about Completing the Square
Why does completing the square go wrong when \(a\neq1\)?
Because the constant produced inside the bracket is multiplied by \(a\). For \(2x^2-8x+3\), writing \(2(x-2)^2\) introduces \(2\times4=8\), so the correction is \(-8\), giving \(2(x-2)^2-5\), not \(2(x-2)^2-1\). Factor \(a\) out of the \(x^2\) and \(x\) terms first, complete the square inside, then multiply the correction back by \(a\). Expand your answer to check it matches the original.
Should I give quadratic roots as surds or decimals in the exam?
If the question says exact, in surd form, or leave your answer in terms of a surd, the surd is the answer and a decimal is a different, approximate one. Solve algebraically by factorising, completing the square or using the formula \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\), which is given in the formula list, and keep the \(\pm\). Then use your calculator only to check the decimal value of your exact answer. Round only when the question asks for it.