ICSE Class 9 Maths Chapter 04 Expansions

Unit 3 - Algebra

Expansions

Points To Remember

1. (i) \((a + b)^2 = a^2 + b^2 + 2ab\) (ii) \((a - b)^2 = a^2 + b^2 - 2ab\)

(iii) \((a + b)^2 + (a - b)^2 = 2(a^2 + b^2)\) (iv) \((a + b)^2 - (a - b)^2 = 4ab\)

(v) \((a + b)(a - b) = (a^2 - b^2)\) (vi) \((a + b)^2 = (a - b)^2 + 4ab\)

(vii) \((a - b)^2 = (a + b)^2 - 4ab\)

2. (i) \(\left(a + \frac{1}{a}\right)^2 = a^2 + \frac{1}{a^2} + 2\) (ii) \(\left(a - \frac{1}{a}\right)^2 = a^2 + \frac{1}{a^2} - 2\)

(iii) \(\left(a + \frac{1}{a}\right)\left(a - \frac{1}{a}\right) = \left(a^2 - \frac{1}{a^2}\right)\) (iv) \(\left(a + \frac{1}{a}\right)^2 + \left(a - \frac{1}{a}\right)^2 = 2\left(a^2 + \frac{1}{a^2}\right)\)

(v) \(\left(a + \frac{1}{a}\right)^2 - \left(a - \frac{1}{a}\right)^2 = 4\) (vi) \(\left(a + \frac{1}{a}\right)^2 = \left(a - \frac{1}{a}\right)^2 + 4\)

(vii) \(\left(a - \frac{1}{a}\right)^2 = \left(a + \frac{1}{a}\right)^2 - 4\)

3. \((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)\)

4. (i) \((x + a)(x + b) = x^2 + (a + b)x + ab\)

(ii) \((x + a)(x - b) = x^2 + (a - b)x - ab\)

(iii) \((x - a)(x + b) = x^2 - (a - b)x - ab\)

(iv) \((x - a)(x - b) = x^2 - (a + b)x + ab\)

Note \((x + a)(x + b) = x^2 + (a + b)x + ab = x^2 + \text{(Algebraic sum of 2nd terms)}x + \text{(Product of second terms)}\)

5. \((a + b)^3 = a^3 + b^3 + 3ab(a + b) = a^3 + 3a^2b + 3ab^2 + b^3\) and \(\left(a + \frac{1}{a}\right)^3 = a^3 + \frac{1}{a^3} + 3\left(a + \frac{1}{a}\right)\)

6. \((a - b)^3 = a^3 - b^3 - 3ab(a - b) = a^3 - 3a^2b + 3ab^2 - b^3\) and \(\left(a - \frac{1}{a}\right)^3 = a^3 - \frac{1}{a^3} - 3\left(a - \frac{1}{a}\right)\)

7. If \(a + b + c = 0\), then \(a^3 + b^3 + c^3 = 3abc\)

Teacher's Note

Understanding algebraic expansions is like breaking down a recipe into individual ingredients - each component plays a specific role, and combining them correctly produces the desired result. These formulas are the building blocks for solving complex mathematical problems in physics and engineering.

Exercise 4 (A)

Using the standard formulae, expand each of the following (Q. No. 1 to 13):

Q. 1. (i) \((4a + 9)^2\)

(ii) \((3x + 10y)^2\)

(iii) \((\sqrt{2}m + \sqrt{3}n)^2\)

Sol. (i) \((4a + 9)^2 = (4a)^2 + (9)^2 + 2 \times 4a \times 9 = 16a^2 + 81 + 72a\)

(ii) \((3x + 10y)^2 = (3x)^2 + (10y)^2 + 2 \times 3x \times 10y = 9x^2 + 100y^2 + 60xy\)

(iii) \((\sqrt{2}m + \sqrt{3}n)^2 = (\sqrt{2}m)^2 + (\sqrt{3}n)^2 + 2 \times \sqrt{2}m \times \sqrt{3}n = 2m^2 + 3n^2 + 2\sqrt{6}mn\) Ans.

Q. 2. (i) \((2a^2 + 3b)^2\)

(ii) \((3x^2y + z)^2\)

(iii) \(\left(2x + \frac{1}{3x}\right)^2\)

Sol. (i) \((2a^2 + 3b)^2 = (2a^2)^2 + (3b)^2 + 2 \times 2a^2 \times 3b = 4a^4 + 9b^2 + 12a^2b\)

(ii) \((3x^2y + z)^2 = (3x^2y)^2 + (z)^2 + 2 \times 3x^2y \times z = 9x^4y^2 + z^2 + 6x^2yz\)

(iii) \(\left(2x + \frac{1}{3x}\right)^2 = (2x)^2 + \left(\frac{1}{3x}\right)^2 + 2 \times 2x \times \frac{1}{3x} = 4x^2 + \frac{1}{9x^2} + \frac{4}{3}\) Ans.

Q. 3. (i) \(\left(\frac{2}{5}x + \frac{5}{6}y\right)^2\) (ii) \(\left(\frac{x}{3} + \frac{6}{x}\right)^2\)

(iii) \(\left(6 + \frac{5}{x}\right)^2\)

Sol. (i) \(\left(\frac{2}{5}x + \frac{5}{6}y\right)^2 = \left(\frac{2}{5}x\right)^2 + \left(\frac{5}{6}y\right)^2 + 2 \times \frac{2}{5}x \times \frac{5}{6}y = \frac{4}{25}x^2 + \frac{25}{36}y^2 + \frac{2}{3}xy\)

(ii) \(\left(\frac{x}{3} + \frac{6}{x}\right)^2 = \left(\frac{x}{3}\right)^2 + \left(\frac{6}{x}\right)^2 + 2 \times \frac{x}{3} \times \frac{6}{x} = \frac{x^2}{9} + \frac{36}{x^2} + 4\)

(iii) \(\left(6 + \frac{5}{x}\right)^2 = (6)^2 + \left(\frac{5}{x}\right)^2 + 2 \times 6 \times \frac{5}{x} = 36 + \frac{25}{x^2} + \frac{60}{x}\) Ans.

Q. 4. (i) \((5x - 3y)^2\)

(ii) \((3a - 7b)^2\)

(iii) \(\left(\frac{1}{2}x - \frac{3}{2}y\right)^2\)

Sol. (i) \((5x - 3y)^2 = (5x)^2 + (3y)^2 - 2 \times 5x \times 3y = 25x^2 + 9y^2 - 30xy\)

(ii) \((3a - 7b)^2 = (3a)^2 + (7b)^2 - 2 \times 3a \times 7b = 9a^2 + 49b^2 - 42ab\)

(iii) \(\left(\frac{1}{2}x - \frac{3}{2}y\right)^2 = \left(\frac{1}{2}x\right)^2 + \left(\frac{3}{2}y\right)^2 - 2 \times \frac{1}{2}x \times \frac{3}{2}y = \frac{1}{4}x^2 + \frac{9}{4}y^2 - \frac{3}{2}xy\) Ans.

Q. 5. (i) \(\left(a^2 - \frac{b}{2}\right)^2\) (ii) \(\left(\frac{3a}{2b} - \frac{2b}{3a}\right)^2\)

(iii) \(\left(5x - \frac{2}{3x}\right)^2\)

Q. 6. (i) \((a + 2b + 3c)^2\)

(ii) \((3x + 5y - 2z)^2\)

(iii) \((2x - 3y + 7z)^2\)

Sol. (i) \((a + 2b + 3c)^2 = (a)^2 + (2b)^2 + (3c)^2 + 2 \times a \times 2b + 2 \times 2b \times 3c + 2 \times 3c \times a = a^2 + 4b^2 + 9c^2 + 4ab + 12bc + 6ca\)

(ii) \((3x + 5y - 2z)^2 = (3x)^2 + (5y)^2 + (2z)^2 + 2 \times 3x \times 5y - 2 \times 5y \times 2z - 2 \times 2z \times 3x = 9x^2 + 25y^2 + 4z^2 + 30xy - 20yz - 12zx\)

(iii) \((2x - 3y + 7z)^2 = (2x)^2 + (3y)^2 + (7z)^2 - 2 \times 2x \times 3y - 2 \times 3y \times 7z + 2 \times 7z \times 2x = 4x^2 + 9y^2 + 49z^2 - 12xy - 42yz + 28zx\) Ans.

Q. 7. (i) \((6 - 2y + 4z)^2\)

(ii) \((4x - 3y + z)^2\)

(iii) \((7 - 2x - 3y)^2\)

Sol. (i) \((6 - 2y + 4z)^2 = (6)^2 + (2y)^2 + (4z)^2 - 2 \times 6 \times 2y - 2 \times 2y \times 4z + 2 \times 4z \times 6 = 36 + 4y^2 + 16z^2 - 24y - 16yz + 48z\)

(ii) \((4x - 3y + z)^2 = (4x)^2 + (3y)^2 + (z)^2 - 2 \times 4x \times 3y - 2 \times 3y \times z + 2 \times z \times 4x = 16x^2 + 9y^2 + z^2 - 24xy - 6yz + 8zx\)

(iii) \((7 - 2x - 3y)^2 = (7)^2 + (2x)^2 + (3y)^2 - 2 \times 7 \times 2x + 2 \times 2x \times 3y - 2 \times 3y \times 7 = 49 + 4x^2 + 9y^2 - 28x + 12xy - 42y\) Ans.

Q. 8. (i) \(\left(\frac{a}{2} + \frac{b}{3} + \frac{c}{4}\right)^2\)

(ii) \(\left(\frac{2x}{3} + \frac{3}{2y} - 2\right)^2\)

(iii) \(\left(2x + \frac{3}{x} - 1\right)^2\)

Sol. (i) \(\left(\frac{a}{2} + \frac{b}{3} + \frac{c}{4}\right)^2 = \left(\frac{a}{2}\right)^2 + \left(\frac{b}{3}\right)^2 + \left(\frac{c}{4}\right)^2 + 2 \times \frac{a}{2} \times \frac{b}{3} + 2 \times \frac{b}{3} \times \frac{c}{4} + 2 \times \frac{c}{4} \times \frac{a}{2} = \frac{a^2}{4} + \frac{b^2}{9} + \frac{c^2}{16} + \frac{1}{3}ab + \frac{1}{6}bc + \frac{1}{4}ca\)

(ii) \(\left(\frac{2x}{3} + \frac{3}{2y} - 2\right)^2 = \left(\frac{2x}{3}\right)^2 + \left(\frac{3}{2y}\right)^2 + (2)^2 + 2 \times \frac{2x}{3} \times \frac{3}{2y} - 2 \times \frac{3}{2y} \times 2 - 2 \times 2 \times \frac{2x}{3}\)

Teacher's Note

When you wrap a gift box with multiple items inside, you must account for every dimension and their interactions - just like in these three-term expansions where each pair of terms contributes to the final result.

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