Maharashtra Board Class 7 Maths Chapter 14 Set 53 Algebraic Formulae Expansion of Squares Solutions

Question 1. Factorize the following expressions:
(i) p² - q²
(ii) 4x² - 25y²
(iii) y² - 4
(iv) p² - \( \frac{1}{25} \)
(v) 9x² - \( \frac{1}{16} \)y²
(vi) x² - \( \frac{1}{x^2} \)
(vii) a²b - ab
(viii) 4x²y - 6x²
(ix) \( \frac{1}{2} \)y² - 8z²
(x) 2x² - 8y²
Answer:
(i) p² - q²
Here, a = p, b = q

\( \implies \) p² - q² = (p + q)(p - q)
....[(a² - b²) = (a + b)(a - b)]
(ii) 4x² - 25y²
= (2x)² - (5y)²
Here, a = 2x, b = 5y

\( \implies \) (2x)² - (5y)² = (2x + 5y)(2x - 5y)
....[(a² - b²) = (a + b)(a - b)]
(iii) y² - 4
= y² - 2²
Here, a = y, b = 2

\( \implies \) y² - 2² = (y + 2)(y - 2)
....[(a² - b²) = (a + b)(a - b)]
(iv) p² - \( \frac{1}{25} \)
Here a = p, b = \( \frac{1}{5} \)

\( \implies \) p² - \( (\frac{1}{5})^2 \) = \( (p + \frac{1}{5}) (p - \frac{1}{5}) \)
....[(a² - b²) = (a + b)(a - b)]
(v) 9x² - \( \frac{1}{16} \)y²
Here a = 3x, b = \( \frac{1}{4} \)y

\( \implies \) (3x)² - \( (\frac{1}{4}y)^2 \) = \( (3x + \frac{1}{4}y) (3x - \frac{1}{4}y) \)
....[(a² - b²) = (a + b)(a - b)]
(vi) x² - \( \frac{1}{x^2} \)
Here a = x, b = \( \frac{1}{x} \)

\( \implies \) x² - \( (\frac{1}{x})^2 \) = \( (x + \frac{1}{x})(x - \frac{1}{x}) \)
....[(a² - b²) = (a + b)(a - b)]
(vii) a²b - ab
= a (ab - b)
= ab (a - 1)
(viii) 4x²y - 6x²
= 2 (2x²y - 3x²)
= 2x² (2y - 3)
(ix) \( \frac{1}{2} \)y² - 8z²
= \( \frac{1}{2} \)y² - \( \frac{2 \times 8}{2} \)z²
= \( \frac{1}{2} \)y² - \( \frac{16}{2} \)z²
= \( \frac{1}{2} \)(y² - 16z²)
= \( \frac{1}{2} \)[y² - (4z)²]
= \( \frac{1}{2} \)[(y + 4z)(y - 4z)]
(x) 2x² - 8y²
= 2 (x² - 4y²)
= 2 [x² - (2y)²]
= 2(x + 2y)(x - 2y)
....[(a² - b²) = (a + b)(a - b)]
In simple words: This question asks to factorize algebraic expressions by identifying common factors or applying the difference of squares formula, \( (a^2 - b^2) = (a - b)(a + b) \), to simplify each given expression into its factors.

🎯 Exam Tip: Mastering the difference of squares formula and identifying common factors is crucial for scoring well in factorization problems. Show each step clearly, especially the identification of 'a' and 'b' terms.