Completing the Square | CAIE 9709 Pure Mathematics 1 Notes

Pure Mathematics 1

Mathematics9709

Why it is called completing the square

$x^2+6x$ makes an $x$ -square and two strips of width $3$ . A $3\times3$ corner completes the larger square.

Swipe left or right to see the whole diagram →

x² + 6x is a square of side x plus two strips of width 3. Adding the small 3 × 3 corner completes a square of side x + 3, so you must subtract that corner again: x² + 6x = (x + 3)² − 9.

x^2+6x=(x+3)^2-9

Add the missing corner inside the square, then subtract it outside.

Halve b, square it, subtract it

Here $b$ is the signed coefficient of $x$ .

x^2+bx=\left(x+\tfrac{b}{2}\right)^2-\left(\tfrac{b}{2}\right)^2

halve bsquare the halfsubtract that squareexpand to check

Worked example

Express $x^2+4x+2$ in the form $(x+a)^2+b$ (9709/12/M/J/25 Q11(a))

Half of $4$ is $2$ .
$=(x+2)^2-4+2$

Answer

(x+2)^2-2

Common mistake

For

x^2-10x+7

, half of

-10

-5

, but

(-5)^2=25

. The result is

(x-5)^2-18

Your turn9709/11/M/J/22 Q1(a)

Express $x^2-8x+11$ in the form $(x+p)^2+q$ .

Show worked answer

=(x-4)^2-16+11

Answer

(x-4)^2-5

When a is not 1

Here $a$ is the coefficient of $x^2$ . Factor it out of the $x^2$ and $x$ terms, complete the square inside the bracket, then distribute $a$ again.

Worked example

Express $2x^2+4x+10$ in the form $a(x+b)^2+c$ (9709/13/O/N/25 Q10(a)(i))

$2(x^2+2x)+10$
$2\big[(x+1)^2-1\big]+10$
$2(x+1)^2-2+10$

Answer

2(x+1)^2+8

Common mistake

The outside factor multiplies every term: in

2[(x-3)^2-9]

-9

becomes

-18

Worked example

Express $1-6x-x^2$ in the form $a-(x+b)^2$ (9709/11/O/N/25 Q4(a))

1-6x-x^2=-(x^2+6x)+1

=-\big[(x+3)^2-9\big]+1

Answer

10-(x+3)^2

Why it is called completing the square

$x^2+6x$ makes an $x$ -square and two strips of width $3$ . A $3\times3$ corner completes the larger square.

Swipe left or right to see the whole diagram →

x² + 6x is a square of side x plus two strips of width 3. Adding the small 3 × 3 corner completes a square of side x + 3, so you must subtract that corner again: x² + 6x = (x + 3)² − 9.

x^2+6x=(x+3)^2-9

Add the missing corner inside the square, then subtract it outside.

Halve b, square it, subtract it

Here $b$ is the signed coefficient of $x$ .

x^2+bx=\left(x+\tfrac{b}{2}\right)^2-\left(\tfrac{b}{2}\right)^2

halve bsquare the halfsubtract that squareexpand to check

Worked example

Express $x^2+4x+2$ in the form $(x+a)^2+b$ (9709/12/M/J/25 Q11(a))

Half of $4$ is $2$ .
$=(x+2)^2-4+2$

Answer

(x+2)^2-2

Common mistake

For

x^2-10x+7

, half of

-10

-5

, but

(-5)^2=25

. The result is

(x-5)^2-18

Your turn9709/11/M/J/22 Q1(a)

Express $x^2-8x+11$ in the form $(x+p)^2+q$ .

Show worked answer

=(x-4)^2-16+11

Answer

(x-4)^2-5

When a is not 1

Here $a$ is the coefficient of $x^2$ . Factor it out of the $x^2$ and $x$ terms, complete the square inside the bracket, then distribute $a$ again.

Worked example

Express $2x^2+4x+10$ in the form $a(x+b)^2+c$ (9709/13/O/N/25 Q10(a)(i))

$2(x^2+2x)+10$
$2\big[(x+1)^2-1\big]+10$
$2(x+1)^2-2+10$

Answer

2(x+1)^2+8

Common mistake

The outside factor multiplies every term: in

2[(x-3)^2-9]

-9

becomes

-18

Worked example

Express $1-6x-x^2$ in the form $a-(x+b)^2$ (9709/11/O/N/25 Q4(a))

1-6x-x^2=-(x^2+6x)+1

=-\big[(x+3)^2-9\big]+1

Answer

10-(x+3)^2