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Algebra and Graphs - Algebraic manipulation and factorisation

Grade 11IGCSE

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

🔑Concepts

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Expanding Brackets: Multiplying out single, double, and triple brackets using the distributive law.

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Factorisation: The reverse of expansion. Includes extracting common factors, grouping terms, and factorising quadratic expressions.

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Difference of Two Squares: Identifying and factorising expressions in the form a2−b2a^2 - b^2.

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Algebraic Fractions: Simplifying, adding, subtracting, multiplying, and dividing fractions containing variables by finding common denominators and cancelling factors.

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Changing the Subject of a Formula: Rearranging an equation to isolate a specific variable, involving techniques like cross-multiplication and factorisation when the variable appears multiple times.

📐Formulae

Difference of Two Squares: a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

Perfect Square (Positive): (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

Perfect Square (Negative): (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

Quadratic Form: ax2+bx+cax^2 + bx + c factorises to (px+q)(rx+s)(px + q)(rx + s) where pr=apr=a and qs=cqs=c.

Algebraic Fraction Addition: ab±cd=ad±bcbd\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}

💡Examples

Problem 1:

Factorise completely: 6x2−15xy6x^2 - 15xy.

Solution:

3x(2x−5y)3x(2x - 5y)

Explanation:

Identify the Highest Common Factor (HCF) of the numerical coefficients (3) and the variables (xx). Divide each term by 3x3x and place the result inside brackets.

Problem 2:

Simplify the algebraic fraction: x2−92x2+5x−3\frac{x^2 - 9}{2x^2 + 5x - 3}.

Solution:

x−32x−1\frac{x - 3}{2x - 1}

Explanation:

Factorise the numerator using the Difference of Two Squares: (x−3)(x+3)(x-3)(x+3). Factorise the denominator: (2x−1)(x+3)(2x-1)(x+3). Cancel the common factor (x+3)(x+3) from both the top and bottom.

Problem 3:

Make xx the subject of the formula: y=2x+1x−3y = \frac{2x + 1}{x - 3}.

Solution:

x=3y+1y−2x = \frac{3y + 1}{y - 2}

Explanation:

  1. Multiply both sides by (x−3)(x-3) to get y(x−3)=2x+1y(x-3) = 2x+1. 2. Expand: yx−3y=2x+1yx - 3y = 2x + 1. 3. Move all xx terms to one side: yx−2x=3y+1yx - 2x = 3y + 1. 4. Factorise xx: x(y−2)=3y+1x(y - 2) = 3y + 1. 5. Divide by (y−2)(y-2) to isolate xx.

Problem 4:

Factorise by grouping: ax−ay+bx−byax - ay + bx - by.

Solution:

(a+b)(x−y)(a + b)(x - y)

Explanation:

Group the first two terms a(x−y)a(x-y) and the last two terms b(x−y)b(x-y). Since (x−y)(x-y) is common to both groups, factorise it out to get (a+b)(x−y)(a+b)(x-y).