Learn the method, study an example, then explain your reasoning in an exam-style problem. Work on paper before opening the solution and compare each step with the marking guidance.
Completing the square and range
Transform a quadratic and interpret its vertex, range and intersections.
Notes
Since \((x+h)^2=x^2+2hx+h^2\), half the coefficient of \(x\) belongs inside the bracket. For \(a\ne0\),
Factor out \(a\) from the first two terms before completing the square. In \(a(x-h)^2+k\), the vertex is \((h,k)\). It is a minimum if \(a>0\) and a maximum if \(a<0\).
On a restricted domain, check whether the vertex is included and evaluate the endpoints. Solving an equation requires both square-root signs.
Common mistakes
Forgetting to multiply the compensation term by a.
Reading the sign of the vertex incorrectly.
Using the unrestricted minimum on a domain that excludes the vertex.
Three worked examples
Example 1
Write \(x^2+8x+11\) in completed-square form.
Show the reasoning
\(x^2+8x+11=(x+4)^2-16+11=(x+4)^2-5\). Its minimum is \(-5\) at \(x=-4\).
Example 2
Find the maximum of \(-3x^2+12x-5\).
Show the reasoning
\(-3(x^2-4x)-5=-3(x-2)^2+7\). The square is nonnegative, so the maximum is \(7\) at \(x=2\).
Example 3
Find the range of \(2(x-1)^2+3\) for \(2\le x\le4\).
Show the reasoning
The vertex \(x=1\) is outside the domain. The function increases here; endpoint values are \(5\) and \(21\). The range is \(5\le y\le21\).