Sec 3 · G2 (N(A))

Algebraic manipulation

Practise algebraic manipulation with step-by-step help. Free account.

1. Expanding

Each term × each term

\displaystyle (a + b)(c + d) = ac + ad + bc + bd

e.g.

\displaystyle (x + 3)(2x - 5)

\displaystyle = 2x^{2} - 5x + 6x - 15

\displaystyle = 2x^{2} + x - 15

Perfect square

\displaystyle (a + b)^{2} = a^{2} + 2ab + b^{2}

e.g.

\displaystyle (x + 5)^{2}

\displaystyle = x^{2} + 10x + 25

Perfect square

\displaystyle (a - b)^{2} = a^{2} - 2ab + b^{2}

e.g.

\displaystyle (2x - 3)^{2}

\displaystyle = 4x^{2} - 12x + 9

Difference of two squares

\displaystyle (a + b)(a - b) = a^{2} - b^{2}

e.g.

\displaystyle (3x + 2)(3x - 2)

\displaystyle = 9x^{2} - 4

📖 Worked example

Expand and simplify $(3x + 4)(2x - 1)$ .

2x

-1

3x

+4

Step 1 of 5

Set up the box

Write one bracket down the side (

3x

and

+4

) and the other across the top (

2x

and

-1

Practise this →

2. Factorising

Common factor

\displaystyle ab + ac = a(b + c)

e.g.

\displaystyle 6x^{2} - 9x

\displaystyle = 3x(2x - 3)

x² + bx + c

\displaystyle x^{2} + bx + c = (x + m)(x + n)

e.g.

\displaystyle x^{2} - x - 12

\displaystyle = (x - 4)(x + 3)

$m + n = b$
$m \times n = c$

Difference of two squares

\displaystyle a^{2} - b^{2} = (a + b)(a - b)

e.g.

\displaystyle 4x^{2} - 25

\displaystyle = (2x + 5)(2x - 5)

Perfect square

\displaystyle a^{2} + 2ab + b^{2} = (a + b)^{2}

e.g.

\displaystyle x^{2} + 6x + 9

\displaystyle = (x + 3)^{2}

ax² + bx + c (cross method)

\displaystyle ax^{2} + bx + c = (px + q)(rx + s)

e.g.

\displaystyle 6x^{2} + x - 2

\displaystyle = (3x + 2)(2x - 1)

$pr = a$ , $qs = c$
$ps + qr = b$

Grouping

\displaystyle ax + ay + bx + by = (a + b)(x + y)

e.g.

\displaystyle 6xy - 4x + 3y - 2

\displaystyle = 2x(3y - 2) + (3y - 2)

\displaystyle = (3y - 2)(2x + 1)

📖 Worked example

Factorise $x^{2} + 3x - 10$ .

x^{2}

-10

Step 1 of 6

Set up the box

Draw a 2 × 2 box. Put

x^{2}

in the top-left and

-10

in the bottom-right.

📖 Worked example

Factorise $6x^{2} + x - 2$ .

6x^{2}

-2

Step 1 of 6

Set up the box

Draw a 2 × 2 box. Put

6x^{2}

in the top-left and

-2

in the bottom-right.

Practise this →

3. Algebraic fractions

Simplify

\displaystyle \dfrac{ac}{bc} = \dfrac{a}{b}

e.g.

\displaystyle \dfrac{x^{2} - 9}{x^{2} + x - 6}

\displaystyle = \dfrac{(x + 3)(x - 3)}{(x + 3)(x - 2)}

\displaystyle = \dfrac{x - 3}{x - 2}

Factorise, then cancel whole brackets

Add or subtract

\displaystyle \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}

e.g.

\displaystyle \dfrac{2}{x + 1} + \dfrac{3}{x - 2}

\displaystyle = \dfrac{2(x - 2) + 3(x + 1)}{(x + 1)(x - 2)}

\displaystyle = \dfrac{5x - 1}{(x + 1)(x - 2)}

Multiply

\displaystyle \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}

e.g.

\displaystyle \dfrac{x^{2} - 4}{x + 3} \times \dfrac{x + 3}{x - 2}

\displaystyle = \dfrac{(x + 2)(x - 2)}{x + 3} \times \dfrac{x + 3}{x - 2}

\displaystyle = x + 2

Divide

\displaystyle \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}

e.g.

\displaystyle \dfrac{x + 1}{x} \div \dfrac{x + 1}{x^{2}}

\displaystyle = \dfrac{x + 1}{x} \times \dfrac{x^{2}}{x + 1}

\displaystyle = x

📖 Worked example

Express $\dfrac{2}{x - 1} - \dfrac{3}{x + 1}$ as a single fraction in its simplest form.

Step 1 of 3

Use a common denominator

\dfrac{2}{x - 1} - \dfrac{3}{x + 1} = \dfrac{2(x + 1) - 3(x - 1)}{(x - 1)(x + 1)}

📖 Worked example

Simplify $\dfrac{x^{2} - 6x + 8}{x^{2} + 5x + 4} \div \dfrac{x - 2}{x + 4}$ .

Step 1 of 3

Dividing: flip the second fraction and multiply

\dfrac{x^{2} - 6x + 8}{x^{2} + 5x + 4} \div \dfrac{x - 2}{x + 4}

= \dfrac{x^{2} - 6x + 8}{x^{2} + 5x + 4} \times \dfrac{x + 4}{x - 2}

Practise this →

4. Changing the subject

Done to $x$	Undo it	Example
$+\,a$	$-\,a$	$y = x + 4 \;\to\; x = y - 4$
$\times\, a$	$\div\, a$	$y = 3x \;\to\; x = \dfrac{y}{3}$
$\div\, a$	$\times\, a$	$y = \dfrac{x}{5} \;\to\; x = 5y$
square	$\pm\sqrt{\phantom{x}}$	$y = x^{2} \;\to\; x = \pm\sqrt{y}$
$\sqrt{\phantom{x}}$	square	$y = \sqrt{x} \;\to\; x = y^{2}$

Undo in reverse order

\displaystyle y = \dfrac{3x + 2}{5}

e.g.

\displaystyle y = \dfrac{3x + 2}{5}

\displaystyle 5y = 3x + 2

\displaystyle 5y - 2 = 3x

\displaystyle x = \dfrac{5y - 2}{3}

x appears twice: collect, then factorise

\displaystyle y = \dfrac{2x + 1}{x - 3}

e.g.

\displaystyle y = \dfrac{2x + 1}{x - 3}

\displaystyle y(x - 3) = 2x + 1

\displaystyle xy - 3y = 2x + 1

\displaystyle xy - 2x = 3y + 1

\displaystyle x(y - 2) = 3y + 1

\displaystyle x = \dfrac{3y + 1}{y - 2}

📖 Worked example

Make $x$ the subject of $y = 3x^{2} - 3$ .

Step 1 of 2

Get x² on its own

y + 3 = 3x^{2}

x^{2} = \dfrac{y + 3}{3}

📖 Worked example

Make $x$ the subject of $y = \sqrt{4x + 4}$ .

Step 1 of 2

Remove the square root

y^{2} = 4x + 4

Practise this →

Try a question: Algebraic manipulation

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise algebraic manipulation with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free

Sec 3 · G2 (N(A))

Algebraic manipulation

Practise algebraic manipulation with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Expanding

Each term × each term

\displaystyle (a + b)(c + d) = ac + ad + bc + bd

e.g.

\displaystyle (x + 3)(2x - 5)

\displaystyle = 2x^{2} - 5x + 6x - 15

\displaystyle = 2x^{2} + x - 15

Perfect square

\displaystyle (a + b)^{2} = a^{2} + 2ab + b^{2}

e.g.

\displaystyle (x + 5)^{2}

\displaystyle = x^{2} + 10x + 25

Perfect square

\displaystyle (a - b)^{2} = a^{2} - 2ab + b^{2}

e.g.

\displaystyle (2x - 3)^{2}

\displaystyle = 4x^{2} - 12x + 9

Difference of two squares

\displaystyle (a + b)(a - b) = a^{2} - b^{2}

e.g.

\displaystyle (3x + 2)(3x - 2)

\displaystyle = 9x^{2} - 4

📖 Worked example

Expand and simplify $(3x + 4)(2x - 1)$ .

2x

-1

3x

+4

Step 1 of 5

Set up the box

Write one bracket down the side (

3x

and

+4

) and the other across the top (

2x

and

-1

Practise this →

2. Factorising

Common factor

\displaystyle ab + ac = a(b + c)

e.g.

\displaystyle 6x^{2} - 9x

\displaystyle = 3x(2x - 3)

x² + bx + c

\displaystyle x^{2} + bx + c = (x + m)(x + n)

e.g.

\displaystyle x^{2} - x - 12

\displaystyle = (x - 4)(x + 3)

$m + n = b$
$m \times n = c$

Difference of two squares

\displaystyle a^{2} - b^{2} = (a + b)(a - b)

e.g.

\displaystyle 4x^{2} - 25

\displaystyle = (2x + 5)(2x - 5)

Perfect square

\displaystyle a^{2} + 2ab + b^{2} = (a + b)^{2}

e.g.

\displaystyle x^{2} + 6x + 9

\displaystyle = (x + 3)^{2}

ax² + bx + c (cross method)

\displaystyle ax^{2} + bx + c = (px + q)(rx + s)

e.g.

\displaystyle 6x^{2} + x - 2

\displaystyle = (3x + 2)(2x - 1)

$pr = a$ , $qs = c$
$ps + qr = b$

Grouping

\displaystyle ax + ay + bx + by = (a + b)(x + y)

e.g.

\displaystyle 6xy - 4x + 3y - 2

\displaystyle = 2x(3y - 2) + (3y - 2)

\displaystyle = (3y - 2)(2x + 1)

📖 Worked example

Factorise $x^{2} + 3x - 10$ .

x^{2}

-10

Step 1 of 6

Set up the box

Draw a 2 × 2 box. Put

x^{2}

in the top-left and

-10

in the bottom-right.

📖 Worked example

Factorise $6x^{2} + x - 2$ .

6x^{2}

-2

Step 1 of 6

Set up the box

Draw a 2 × 2 box. Put

6x^{2}

in the top-left and

-2

in the bottom-right.

Practise this →

3. Algebraic fractions

Simplify

\displaystyle \dfrac{ac}{bc} = \dfrac{a}{b}

e.g.

\displaystyle \dfrac{x^{2} - 9}{x^{2} + x - 6}

\displaystyle = \dfrac{(x + 3)(x - 3)}{(x + 3)(x - 2)}

\displaystyle = \dfrac{x - 3}{x - 2}

Factorise, then cancel whole brackets

Add or subtract

\displaystyle \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}

e.g.

\displaystyle \dfrac{2}{x + 1} + \dfrac{3}{x - 2}

\displaystyle = \dfrac{2(x - 2) + 3(x + 1)}{(x + 1)(x - 2)}

\displaystyle = \dfrac{5x - 1}{(x + 1)(x - 2)}

Multiply

\displaystyle \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}

e.g.

\displaystyle \dfrac{x^{2} - 4}{x + 3} \times \dfrac{x + 3}{x - 2}

\displaystyle = \dfrac{(x + 2)(x - 2)}{x + 3} \times \dfrac{x + 3}{x - 2}

\displaystyle = x + 2

Divide

\displaystyle \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}

e.g.

\displaystyle \dfrac{x + 1}{x} \div \dfrac{x + 1}{x^{2}}

\displaystyle = \dfrac{x + 1}{x} \times \dfrac{x^{2}}{x + 1}

\displaystyle = x

📖 Worked example

Express $\dfrac{2}{x - 1} - \dfrac{3}{x + 1}$ as a single fraction in its simplest form.

Step 1 of 3

Use a common denominator

\dfrac{2}{x - 1} - \dfrac{3}{x + 1} = \dfrac{2(x + 1) - 3(x - 1)}{(x - 1)(x + 1)}

📖 Worked example

Simplify $\dfrac{x^{2} - 6x + 8}{x^{2} + 5x + 4} \div \dfrac{x - 2}{x + 4}$ .

Step 1 of 3

Dividing: flip the second fraction and multiply

\dfrac{x^{2} - 6x + 8}{x^{2} + 5x + 4} \div \dfrac{x - 2}{x + 4}

= \dfrac{x^{2} - 6x + 8}{x^{2} + 5x + 4} \times \dfrac{x + 4}{x - 2}

Practise this →

4. Changing the subject

Done to $x$	Undo it	Example
$+\,a$	$-\,a$	$y = x + 4 \;\to\; x = y - 4$
$\times\, a$	$\div\, a$	$y = 3x \;\to\; x = \dfrac{y}{3}$
$\div\, a$	$\times\, a$	$y = \dfrac{x}{5} \;\to\; x = 5y$
square	$\pm\sqrt{\phantom{x}}$	$y = x^{2} \;\to\; x = \pm\sqrt{y}$
$\sqrt{\phantom{x}}$	square	$y = \sqrt{x} \;\to\; x = y^{2}$

Undo in reverse order

\displaystyle y = \dfrac{3x + 2}{5}

e.g.

\displaystyle y = \dfrac{3x + 2}{5}

\displaystyle 5y = 3x + 2

\displaystyle 5y - 2 = 3x

\displaystyle x = \dfrac{5y - 2}{3}

x appears twice: collect, then factorise

\displaystyle y = \dfrac{2x + 1}{x - 3}

e.g.

\displaystyle y = \dfrac{2x + 1}{x - 3}

\displaystyle y(x - 3) = 2x + 1

\displaystyle xy - 3y = 2x + 1

\displaystyle xy - 2x = 3y + 1

\displaystyle x(y - 2) = 3y + 1

\displaystyle x = \dfrac{3y + 1}{y - 2}

📖 Worked example

Make $x$ the subject of $y = 3x^{2} - 3$ .

Step 1 of 2

Get x² on its own

y + 3 = 3x^{2}

x^{2} = \dfrac{y + 3}{3}

📖 Worked example

Make $x$ the subject of $y = \sqrt{4x + 4}$ .

Step 1 of 2

Remove the square root

y^{2} = 4x + 4

Practise this →

Try a question: Algebraic manipulation

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise algebraic manipulation with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free