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Algebra - Algebraic Fractions

Grade 10IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

πŸ”‘Concepts

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Simplifying Algebraic Fractions: Factorize the numerator and denominator completely before canceling common factors.

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Multiplication: Multiply numerators together and denominators together. Simplify the resulting fraction where possible.

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Division: Multiply the first fraction by the reciprocal (the 'flipped' version) of the second fraction.

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Addition and Subtraction: Find a common denominator (usually the Lowest Common Multiple of the denominators) before combining numerators.

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Solving Equations: Clear fractions by multiplying every term by the common denominator or by cross-multiplying if the equation is in the form a/b = c/d.

πŸ“Formulae

abΓ—cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

abΓ·cd=abΓ—dc=adbc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

abΒ±cd=adΒ±bcbd\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}

a2βˆ’b2=(aβˆ’b)(a+b)a^2 - b^2 = (a - b)(a + b) (Difference of two squares used for simplifying)

πŸ’‘Examples

Problem 1:

Simplify x2βˆ’9x2+5x+6\frac{x^2 - 9}{x^2 + 5x + 6}

Solution:

xβˆ’3x+2\frac{x-3}{x+2}

Explanation:

First, factorize both the numerator and the denominator. The numerator is a difference of squares: x2βˆ’9=(xβˆ’3)(x+3)x^2 - 9 = (x-3)(x+3). The denominator is a quadratic: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x+2)(x+3). Cancel the common factor (x+3)(x+3) from both.

Problem 2:

Express as a single fraction: \frac{3}{x+1} - rac{2}{x-2}

Solution:

xβˆ’8(x+1)(xβˆ’2)\frac{x-8}{(x+1)(x-2)}

Explanation:

Find a common denominator, which is (x+1)(xβˆ’2)(x+1)(x-2). Rewrite each fraction: 3(xβˆ’2)(x+1)(xβˆ’2)βˆ’2(x+1)(x+1)(xβˆ’2)\frac{3(x-2)}{(x+1)(x-2)} - \frac{2(x+1)}{(x+1)(x-2)}. Expand the numerators: 3xβˆ’6βˆ’(2x+2)3x - 6 - (2x + 2). Be careful with the minus sign: 3xβˆ’6βˆ’2xβˆ’2=xβˆ’83x - 6 - 2x - 2 = x - 8.

Problem 3:

Solve 4x+32x=11\frac{4}{x} + \frac{3}{2x} = 11

Solution:

x=0.5x = 0.5

Explanation:

The common denominator for the left side is 2x2x. Multiply the first term by 2/22/2 to get 82x+32x=11\frac{8}{2x} + \frac{3}{2x} = 11. This simplifies to 112x=11\frac{11}{2x} = 11. Multiplying both sides by 2x2x gives 11=22x11 = 22x. Dividing by 2222 gives x=11/22=0.5x = 11/22 = 0.5.