krit.club logo

Algebra - Algebraic Manipulation and Factorisation

Grade 10IGCSE

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

🔑Concepts

•

Expansion of algebraic expressions (Distributive law)

•

Factorisation by extracting common factors

•

Factorisation by grouping terms (four-term expressions)

•

Factorising quadratic trinomials of the form ax2+bx+cax^2 + bx + c

•

Difference of two squares

•

Simplifying algebraic fractions by cancelling common factors

•

Addition, subtraction, multiplication, and division of algebraic fractions

•

Changing the subject of a formula, including cases where the subject appears twice

📐Formulae

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

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

ab÷cd=ab×dc=adbc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

💡Examples

Problem 1:

Factorise completely: 12x2y−18xy212x^2y - 18xy^2

Solution:

6xy(2x−3y)6xy(2x - 3y)

Explanation:

Identify the Highest Common Factor (HCF) for the coefficients (6) and the variables (xyxy). Divide each term by 6xy6xy to find the expression inside the bracket.

Problem 2:

Factorise x2−5x−14x^2 - 5x - 14

Solution:

(x−7)(x+2)(x - 7)(x + 2)

Explanation:

Find two numbers that multiply to give −14-14 (the constant term) and add to give −5-5 (the coefficient of xx). Those numbers are −7-7 and +2+2.

Problem 3:

Simplify 2x2−8x2+2x−8\frac{2x^2 - 8}{x^2 + 2x - 8}

Solution:

2(x+2)x+4\frac{2(x + 2)}{x + 4}

Explanation:

Factorise the numerator: 2(x2−4)=2(x−2)(x+2)2(x^2 - 4) = 2(x - 2)(x + 2) using the difference of two squares. Factorise the denominator: (x+4)(x−2)(x + 4)(x - 2). Cancel the common factor (x−2)(x - 2) from both the numerator and denominator.

Problem 4:

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

Solution:

x=2y+3y−1x = \frac{2y + 3}{y - 1}

Explanation:

Multiply both sides by (x−2)(x-2) to get y(x−2)=x+3y(x - 2) = x + 3. Expand the bracket: xy−2y=x+3xy - 2y = x + 3. Move all terms with xx to one side: xy−x=2y+3xy - x = 2y + 3. Factorise xx: x(y−1)=2y+3x(y - 1) = 2y + 3. Divide by (y−1)(y - 1).