Sec 3 · G1 (N(T))

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

📖 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)

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

📖 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.

Practise this →

3. 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}

📖 Worked example

Make $x$ the subject of $y = \dfrac{2x - 4}{5}$ .

Step 1 of 3

Clear the fraction

5y = 2x - 4

Practise this →

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

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

📖 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)

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

📖 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.

Practise this →

3. 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}

📖 Worked example

Make $x$ the subject of $y = \dfrac{2x - 4}{5}$ .

Step 1 of 3

Clear the fraction

5y = 2x - 4

Practise this →