Technique

Range of a quadratic by completing the square

Algebra • Quadratic Equations • Maximum and Minimum Values

Statement

Writing

a t^{2} + b t + c

a (t + \frac{b}{2 a})^{2} + c - \frac{b ^{2}}{4 a}

shows that for

a > 0

the expression has least value

c - \frac{b ^{2}}{4 a}

t = - \frac{b}{2 a}

and no greatest value, and for

a < 0

that same point gives the greatest value; on a restricted interval each extreme is taken either at that point, when it lies in the interval, or at an endpoint.

How to use it

1Substitute for a repeated sub-expression first, for instance $t = cos^{2} θ$ , so that what remains is a quadratic in one variable with a known range.
2Check whether the vertex lies inside the allowed interval; if it does not, the quadratic is monotonic there and both extremes sit at the ends.
3Say whether each bound is attained: an open end gives a value that is approached but never reached, which is what decides between $<$ and $\leq$ .

Worked example

Find the least and the greatest value of

t^{2} - 4 t + 7

for

0 \leq t \leq 3

Step-by-step

1
$t^{2} - 4 t + 7 = (t - 2)^{2} + 3$
Why: Completing the square exposes the vertex and the least value at once.
2
$t = 2 lies in [0, 3] \Rightarrow least value = 3$
Why: The square is zero there and positive everywhere else.
3
$f (0) = 7, f (3) = 4 \Rightarrow greatest value = 7$
Why: An upward parabola takes its largest value on an interval at one of the endpoints.

Answer

Least value

3

, greatest value

7

Watch out

The vertex is an extreme value only when it lies in the allowed interval. For

a < 0

the roles of least and greatest are exchanged, and a half-open interval can leave one bound approached but never attained.

Builds on

Square of a sum and of a difference

Also known as

completing the square, vertex of a parabola, minimum value of a quadratic, range of a quadratic, maximum of ax^2+bx+c

Appears in past papers

Statement

Worked example

Appears in past papers

2019 (II)

2018 (II)

2016 (II)