Sec 2 · G2 (N(A))

Algebraic fractions and formulae

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1. Simplifying algebraic fractions

Cancel common factors, never single terms

\displaystyle \dfrac{6x^{2}y}{9xy^{3}}

e.g.

\displaystyle \dfrac{6x^{2}y}{9xy^{3}}

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

Factorise first, then cancel the bracket

\displaystyle \dfrac{x^{2} - 9}{x + 3}

e.g.

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

\displaystyle = x - 3

📖 Worked example

Simplify $\dfrac{x^{2} - 11x + 30}{x - 5}$ .

Step 1 of 2

Factorise every quadratic

\dfrac{x^{2} - 11x + 30}{x - 5} = \dfrac{(x - 5)(x - 6)}{(x - 5)}

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2. Multiplying, dividing, adding and subtracting

Divide: multiply by the reciprocal

\displaystyle \dfrac{4a}{5b} \div \dfrac{2a^{2}}{15}

e.g.

\displaystyle \dfrac{4a}{5b} \times \dfrac{15}{2a^{2}}

\displaystyle = \dfrac{6}{ab}

Add: a common denominator

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

e.g.

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

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

📖 Worked example

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

Step 1 of 3

Use a common denominator

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

Practise this →

3. Formulae

Substitute: v² = u² + 2as with u = 3, a = 2, s = 4

\displaystyle v^{2} = u^{2} + 2as

e.g.

\displaystyle 3^{2} + 2(2)(4)

\displaystyle = 25

$v^{2} = 25$ , so $v = 5$ or $v = -5$

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1. Simplifying algebraic fractions

Cancel common factors, never single terms

\displaystyle \dfrac{6x^{2}y}{9xy^{3}}

e.g.

\displaystyle \dfrac{6x^{2}y}{9xy^{3}}

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

Factorise first, then cancel the bracket

\displaystyle \dfrac{x^{2} - 9}{x + 3}

e.g.

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

\displaystyle = x - 3

📖 Worked example

Simplify $\dfrac{x^{2} - 11x + 30}{x - 5}$ .

Step 1 of 2

Factorise every quadratic

\dfrac{x^{2} - 11x + 30}{x - 5} = \dfrac{(x - 5)(x - 6)}{(x - 5)}

Practise this →

2. Multiplying, dividing, adding and subtracting

Divide: multiply by the reciprocal

\displaystyle \dfrac{4a}{5b} \div \dfrac{2a^{2}}{15}

e.g.

\displaystyle \dfrac{4a}{5b} \times \dfrac{15}{2a^{2}}

\displaystyle = \dfrac{6}{ab}

Add: a common denominator

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

e.g.

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

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

📖 Worked example

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

Step 1 of 3

Use a common denominator

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

Practise this →