Completed Square Form and Turning Points

Turning points from completed square form: reading (−b, c), completing the square first, sketching with the y-intercept, and shifted curves.

In the exam for Edexcel Further Pure Maths 4PM1, Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, IGCSE Additional Maths 0606 · 32 real past-paper questions below

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Reading the Turning Point

Completed square form shows the turning point with no extra working. The number with x changes sign; the number outside stays.

Formula

y = a (x + b)^{2} + c \Rightarrow turning point (- b, c)

$a$ is the number in front of the bracket, $b$ the number added to x inside it, and $c$ the number on its own outside. If the form is $a (x - b)^{2} + c$ , the turning point is $(b, c)$ .

Worked example

1
$f (x) = 6 - 2 (x + 3)^{2}$ — The curve. The squared bracket has −2 in front of it.
2
$(x + 3)^{2} \geq 0$ — A squared number is never negative. The smallest it can be is 0.
3
$x + 3 = 0, so x = - 3$ — The bracket is 0 when x = −3. The +3 changes sign.
4
$f (- 3) = 6 - 2 (0)^{2} = 6$ — Put x = −3 in. The bracket part vanishes, leaving the number outside.
5
$(- 3, 6)$ — The turning point. Any other x takes something away from 6, so 6 is the greatest value: a maximum.

Rule

If a is positive the turning point is a minimum, and c is the least value of y. If a is negative it is a maximum, and c is the greatest value.

Complete the Square, Then Read

Most questions give the expanded form. Part (a) completes the square — the full method is in the lesson The Quadratic Formula and Completing the Square. Part (b) reads the turning point.

Worked example

1
$4 + 12 x - 2 x^{2}$ — The question. The x² has −2 in front, so take −2 out first.
2
$= - 2 (x^{2} - 6 x) + 4$ — Take −2 out of the x terms only: $- 2 x^{2} \div (- 2) = x^{2}$ and $12 x \div (- 2) = - 6 x$ . The 4 stays outside.
3
$x^{2} - 6 x = (x - 3)^{2} - 9$ — Complete the square inside. Half of −6 is −3, and $(x - 3)^{2}$ makes an extra 9, so take 9 away.
4
$= - 2 [(x - 3)^{2} - 9] + 4$ — Put that back in place of $x^{2} - 6 x$ .
5
$= - 2 (x - 3)^{2} + 18 + 4$ — Open the big bracket. The −2 multiplies both parts: (−2) × (−9) = +18.
6
$= 22 - 2 (x - 3)^{2}$ — 18 + 4 = 22. Written with the number first, as in the form a − b(x − c)².
7
$x - 3 = 0, so x = 3$ — For part (b): the bracket is 0 when x = 3. The −3 changes sign.
8
$A = (3, 22)$ — Then y = 22. It is a maximum, because the −2 makes the curve upside down.

Common mistake

Reading the point as (−3, 22). The bracket is (x − 3), and that is 0 when x = 3.

Key idea

Part (b) is marked from your part (a). If (a) has a slip but you read its turning point correctly, you still get the mark — so always answer (b).

Quick Check: x = −b ÷ 2a

A fast way to find where the turning point is, straight from the expanded form. Use it to check your completed square, or when a question just says “find the turning point”.

Formula

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

For $y = a x^{2} + b x + c$ : $a$ is the number in front of $x^{2}$ and $b$ the number in front of x, each with its sign. This gives the x of the turning point; put it back in the equation for y.

Worked example

1
$y = 5 - 6 x - x^{2}$ — The curve. a = −1 (the x² has a minus) and b = −6.
2
$x = - \frac{( - 6 )}{2 \times ( - 1 )}$ — Put b and a into the formula. Keep each negative in brackets so no sign is lost.
3
$x = \frac{6}{- 2}$ — Top: −(−6) = 6, two minuses make a plus. Bottom: 2 × (−1) = −2.
4
$x = - 3$ — 6 ÷ (−2) = −3. A positive divided by a negative is negative.
5
$y = 5 - 6 (- 3) - (- 3)^{2}$ — Put x = −3 back into the equation to find y.
6
$y = 5 + 18 - 9 = 14$ — −6 × (−3) = +18, and (−3)² = 9, so the last term is −9.
7
$(- 3, 14)$ — The turning point. It is a maximum, because a is negative.

Key idea

Check: completing the square gives 14 − (x + 3)², which turns at (−3, 14) too. But if a question says “use your answer to part (a)”, read the point from your completed square — that is what earns the marks.

From Turning Point to Equation

Some questions give the turning point and ask for the equation, such as “y = x² + bx + c has a minimum at (4, −9). Find b and c.” Work backwards.

Worked example

1
$(4, - 9)$ — The minimum. x = 4 means the bracket is (x − 4), because x − 4 = 0 at x = 4. y = −9 is the number outside.
2
$y = (x - 4)^{2} - 9$ — The completed square form. The x² has nothing in front, so there is no number before the bracket.
3
$(x - 4)^{2} = x^{2} - 8 x + 16$ — Expand the bracket: −4x twice makes −8x, and (−4)² = +16.
4
$y = x^{2} - 8 x + 16 - 9$ — Put the expanded bracket back in. The −9 is still on the end.
5
$y = x^{2} - 8 x + 7$ — 16 − 9 = 7. So b = −8 and c = 7.
6
$4^{2} - 8 (4) + 7 = - 9$ — Check: put x = 4 in. 16 − 32 + 7 = −9, the y of the turning point.

Common mistake

Writing (x + 4)². The bracket must be 0 at x = 4, and only x − 4 is 0 there.

Sketching the Curve

A sketch needs three things: the right shape, the turning point and the y-intercept, each labelled with its coordinates.

Worked example

1
$y = 2 x^{2} + 8 x + 3$ — The curve. The x² term is positive, so the curve is a U shape.
2
$= 2 (x^{2} + 4 x) + 3$ — Take the 2 out of the x terms only. The 3 stays outside.
3
$= 2 [(x + 2)^{2} - 4] + 3$ — Complete the square inside: half of 4 is 2, and (x + 2)² makes an extra 4, so take 4 away.
4
$= 2 (x + 2)^{2} - 5$ — 2 × (−4) = −8, and −8 + 3 = −5.
5
$(- 2, - 5)$ — The turning point: the bracket is 0 at x = −2. A minimum, because the 2 in front is positive.
6
$(0, 3)$ — Put x = 0 for the y-intercept: every x term is 0, leaving 3.
7
$(- 4, 3)$ — The curve is symmetric about x = −2. (0, 3) is 2 to the right of it, so (−4, 3) is 2 to the left.
8
$a U through all three$ — The bottom is below the x-axis, so the curve crosses it twice, both times between x = −4 and x = 0.

Common mistake

Drawing an upside-down U. The x² term is +2x², which is positive, so the curve opens upwards.

A Shifted Curve

If every x in an equation is replaced by (x + 2), the curve moves 2 to the left. The y value of the turning point stays the same.

Worked example

1
$x^{2} - 6 x + 1 = (x - 3)^{2} - 8$ — Part (a): half of −6 is −3, (x − 3)² makes an extra 9, and −9 + 1 = −8.
2
$minimum (3, - 8)$ — The bracket is 0 at x = 3, leaving y = −8.
3
$y = (x + 2)^{2} - 6 (x + 2) + 1$ — The new curve: every x has become x + 2.
4
$y = ((x + 2) - 3)^{2} - 8$ — So make the same swap in the completed square: replace x by x + 2 there too.
5
$y = (x - 1)^{2} - 8$ — Inside the bracket, +2 − 3 = −1.
6
$M = (1, - 8)$ — The bracket is 0 at x = 1. The curve moved 2 to the left (3 → 1); the height −8 did not change.

Key idea

Shortcut: x + k in place of x moves the curve k to the left; x − k moves it k to the right. Translations are covered in full in the lesson Transformations of Graphs.

Common mistake

Moving right. A plus inside the bracket moves the curve to the left.

How Many Solutions?

Some questions ask for which values of k an equation such as x² + 10x + 18 = k has two solutions. Think of y = k as a flat line crossing the curve.

Worked example

1
$x^{2} + 10 x + 18 = (x + 5)^{2} - 7$ — Complete the square: half of 10 is 5, (x + 5)² makes an extra 25, and −25 + 18 = −7.
2
$minimum (- 5, - 7)$ — So the lowest the curve goes is y = −7.
3
$y = k$ — The solutions of the equation are the x values where the flat line y = k meets the curve.
4
$k > - 7$ — Two solutions: a line above the lowest point cuts the U twice.
5
$k = - 7$ — One solution: the line just touches the bottom, at x = −5.
6
$k < - 7$ — No solutions: the line passes under the curve and never meets it.

Key idea

For a maximum (an upside-down U) it flips: two solutions when k is below the greatest value.

Which Method? Read the Question

Find the form first, then read or move.

You see	You do
a(x + b)² + c	turning point (−b, c)
Expanded form	complete the square first, or x = −b ÷ 2a
Turning point given	write the bracket form, then expand
Sketch	shape, turning point, y-intercept
x replaced by x + k	move k to the left
= k has two solutions	k above the minimum (below a maximum)
Solve = 0 exactly	Quadratic Formula

Key idea

Negative in front of the square: a maximum. Positive: a minimum.

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