Completing the Square Explained (Interactive, Step by Step) — GCSE Maths

Q: Why is completing the square useful?

It lets you find the exact vertex without calculus, solve any quadratic (it is how the quadratic formula is derived), sketch parabolas quickly, and integrate certain functions in later study.

Completing the square rewrites a quadratic $ax^2+bx+c$ in the form $a(x+p)^2+q$ . That single rearrangement hands you the turning point, the minimum (or maximum) value and the line of symmetry for free — no calculus required. Here’s the method, step by step, with the geometry that gives it its name.

Why bother?

A quadratic in standard form, $x^2+4x+1$ , hides its most useful features. In completed-square form, $(x+2)^2-3$ , everything is on display: the graph is the basic parabola $x^2$ shifted 2 left and 3 down, so its lowest point is at $(-2,-3)$ . Completing the square is also how the quadratic formula itself is derived, and it’s a tool you’ll reuse in integration and coordinate geometry later.

You’re trading a form that’s easy to expand for one that’s easy to read.

Step by step

Watch the manipulation unfold one line at a time for $x^2+4x+1$ . The key move is in step 3: halve the coefficient of $x$ , then subtract the square of that half to undo what the bracket secretly adds.

InteractiveCompleting the square, line by line

Loading interactive…

Step through the algebra. Each line shows one move and why it keeps the expression equal.

Text description ↓

A step-by-step derivation of completing the square for x² + 4x + 1:

x² + 4x + 1 (start)
(x² + 4x) + 1 — group the x terms
(x + 2)² − 4 + 1 — half of 4 is 2; the bracket adds an extra 2² = 4, so subtract it
(x + 2)² − 3 — simplify the constants

The “square” you’re completing

The name is literal. Picture $x^2+4x$ as an $x\times x$ square plus two $2\times x$ rectangles (splitting the $4x$ as $2x+2x$ along two sides). Those pieces almost form a bigger square of side $x+2$ — but a $2\times 2$ corner is missing. Add that $4$ to “complete” the square into $(x+2)^2$ , and subtract it again to keep the value unchanged. That missing corner is exactly the $(b/2)^2$ you subtract.

What vertex form tells you

Once a quadratic is $a(x+p)^2+q$ , the graph reads off instantly: the turning point is at $(-p,\,q)$ and the line of symmetry is $x=-p$ . Drag the point on $x^2+4x+1$ below and confirm the lowest point sits at $(-2,-3)$ , precisely the numbers in $(x+2)^2-3$ .

InteractiveThe parabola behind the algebra

Loading interactive…

The minimum sits at (−2, −3) — the same −2 and −3 that appear in (x + 2)² − 3.

Text description ↓

An interactive parabola of y = x² + 4x + 1. Its vertex (minimum) is at (−2, −3) and its line of symmetry is x = −2, matching the completed-square form (x + 2)² − 3.

Worked examples

1Worked example — a negative constant

Complete the square for $x^2 - 6x + 5$ . Half of $-6$ is $-3$ , and $(-3)^2 = 9$ :

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

Turning point $(3,-4)$ ; minimum value $-4$ .

2Worked example — when a ≠ 1

For $2x^2+12x+7$ , factor the $2$ out of the $x$ terms first:

2\big(x^2+6x\big)+7 = 2\big((x+3)^2-9\big)+7 = 2(x+3)^2 - 11.

Turning point $(-3,-11)$ . Remember to multiply the $-9$ back by the $2$ .

Common mistakes

Your turn

Write $x^2 + 8x + 3$ in completed-square form and state its minimum point.

Show the answer ↓

Half of 8 is 4, $4^2=16$ : $(x+4)^2 - 16 + 3 = (x+4)^2 - 13$ . Minimum at $(-4,-13)$ .

Frequently asked questions

What does completing the square mean?+

It means rewriting a quadratic ax² + bx + c in the form a(x + p)² + q. This "completed square" form makes the turning point, minimum or maximum value, and line of symmetry immediately readable.

How do you complete the square when the coefficient of x² is 1?+

Halve the coefficient of x to get p, then subtract p² to compensate: x² + bx + c = (x + b/2)² − (b/2)² + c. The bracket squared reproduces the bx term, and you correct the extra p² you introduced.

What is the turning point from completed square form?+

For a(x + p)² + q, the turning point is at (−p, q). If a > 0 it is a minimum; if a < 0 it is a maximum. The line of symmetry is x = −p.

Why is completing the square useful?+

It lets you find the exact vertex without calculus, solve any quadratic (it is how the quadratic formula is derived), sketch parabolas quickly, and integrate certain functions in later study.

Why bother?

A quadratic in standard form,

x^2+4x+1

, hides its most useful features. In completed-square form,

(x+2)^2-3

, everything is on display: the graph is the basic parabola

x^2

shifted 2 left and 3 down, so its lowest point is at

(-2,-3)

. Completing the square is also how the quadratic formula itself is derived, and it’s a tool you’ll reuse in integration and coordinate geometry later.

You’re trading a form that’s easy to expand for one that’s easy to read.

Step by step

Watch the manipulation unfold one line at a time for

x^2+4x+1

. The key move is in step 3: halve the coefficient of

x

, then subtract the square of that half to undo what the bracket secretly adds.

InteractiveCompleting the square, line by line

Loading interactive…

Step through the algebra. Each line shows one move and why it keeps the expression equal.

Text description ↓

A step-by-step derivation of completing the square for x² + 4x + 1:

x² + 4x + 1 (start)
(x² + 4x) + 1 — group the x terms
(x + 2)² − 4 + 1 — half of 4 is 2; the bracket adds an extra 2² = 4, so subtract it
(x + 2)² − 3 — simplify the constants

The “square” you’re completing

The name is literal. Picture

x^2+4x

as an

x\times x

square plus two

2\times x

rectangles (splitting the

4x

2x+2x

along two sides). Those pieces almost form a bigger square of side

x+2

— but a

2\times 2

corner is missing. Add that

4

to “complete” the square into

(x+2)^2

, and subtract it again to keep the value unchanged. That missing corner is exactly the

(b/2)^2

you subtract.

What vertex form tells you

Once a quadratic is

a(x+p)^2+q

, the graph reads off instantly: the turning point is at

(-p,\,q)

and the line of symmetry is

x=-p

. Drag the point on

x^2+4x+1

below and confirm the lowest point sits at

(-2,-3)

, precisely the numbers in

(x+2)^2-3

InteractiveThe parabola behind the algebra

Loading interactive…

The minimum sits at (−2, −3) — the same −2 and −3 that appear in (x + 2)² − 3.

Text description ↓

An interactive parabola of y = x² + 4x + 1. Its vertex (minimum) is at (−2, −3) and its line of symmetry is x = −2, matching the completed-square form (x + 2)² − 3.

Worked examples

1Worked example — a negative constant

Complete the square for $x^2 - 6x + 5$ . Half of $-6$ is $-3$ , and $(-3)^2 = 9$ :

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

Turning point $(3,-4)$ ; minimum value $-4$ .

2Worked example — when a ≠ 1

For $2x^2+12x+7$ , factor the $2$ out of the $x$ terms first:

2\big(x^2+6x\big)+7 = 2\big((x+3)^2-9\big)+7 = 2(x+3)^2 - 11.

Turning point $(-3,-11)$ . Remember to multiply the $-9$ back by the $2$ .

Frequently asked questions

What does completing the square mean?+

It means rewriting a quadratic ax² + bx + c in the form a(x + p)² + q. This "completed square" form makes the turning point, minimum or maximum value, and line of symmetry immediately readable.

How do you complete the square when the coefficient of x² is 1?+

What is the turning point from completed square form?+

For a(x + p)² + q, the turning point is at (−p, q). If a > 0 it is a minimum; if a < 0 it is a maximum. The line of symmetry is x = −p.

Why is completing the square useful?+

It lets you find the exact vertex without calculus, solve any quadratic (it is how the quadratic formula is derived), sketch parabolas quickly, and integrate certain functions in later study.

Completing the Square, Step by Step

Why bother?

Step by step

The “square” you’re completing

What vertex form tells you

Worked examples

Common mistakes

Frequently asked questions

Keep exploring

Want a tutor to walk you through it?

Completing the Square, Step by Step

Why bother?

Step by step

The “square” you’re completing

What vertex form tells you

Worked examples

Common mistakes

Frequently asked questions

Keep exploring

Want a tutor to walk you through it?