Maharashtra Board Class 8 Maths Chapter 5 Expansion Formulae Set 5.4 Solutions

Question 1. Expand:
i. (2p + q + 5)²
ii. (m + 2n + 3r)²
iii. (3x + 4y – 5p)²
iv. (7m – 3n – 4k)²
Answer:
Solution:
i. (2p + q + 5)² = \((2p)^2 + (q)^2 + (5)^2 + 2(2p) (q) + 2(q) (5) + 2(2p) (5)\) ... \([(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac]\) = \(4p^2 + q^2 + 25 + 4pq + 10q + 20p\)
ii. (m + 2n + 3r)² = \((m)^2 + (2n)^2 + (3r)^2 + 2(m) (2n) + 2(2n) (3r) + 2(m) (3r)\) ... \([(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac]\) = \(m^2 + 4n^2 + 9r^2 + 4mn + 12nr + 6mr\)
iii. (3x + 4y – 5p)² = \((3x)^2 + (4y)^2 + (- 5p)^2 + 2(3x) (4y) + 2(4y) (- 5p) + 2(3x) (- 5p)\) ... \([(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac]\) = \(9x^2 + 16y^2 + 25p^2 + 24xy - 40py - 30px\)
iv. (7m – 3n – 4k)² = \((7m)^2 + (- 3n)^2 + (- 4k)^2 + 2(7m) (- 3n) + 2 (- 3n) (- 4k) + 2 (7m) (- 4k)\) ... \([(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac]\) = \(49m^2 + 9n^2 + 16k^2 - 42mn + 24nk - 56km\) In simple words: This question demonstrates the expansion of trinomials using the algebraic identity \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac\). Each part applies this formula to different combinations of terms to get the expanded form.

🎯 Exam Tip: Remember the trinomial expansion formula and be careful with signs when substituting negative terms into the formula.

Question 2. Simplify:
i. (x – 2y + 3)² + (x + 2y – 3)²
ii. (3k – 4r – 2m)² – (3k + 4r – 2m)²
iii. (7a – 6b + 5c)² + (7a + 6b – 5c)²
Answer:
Solution:
i. (x – 2y + 3)² + (x + 2y – 3)² = \([(x)^2 + (- 2y)^2 + (3)^2 + 2 (x) (- 2y) + 2 (- 2y) (3) + 2 (x) (3)] + [(x)^2 + (2y)^2 + (- 3)^2 + 2 (x) (2y) + 2 (2y) (- 3) + 2 (x) (- 3)]\) ... \([(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac]\) = \(x^2 + 4y^2 + 9 - 4xy - 12y + 6x + x^2 + 4y^2 + 9 + 4xy - 12y - 6x\) = \(x^2 + x^2 + 4y^2 + 4y^2 + 9 + 9 - 4xy + 4xy - 12y - 12y + 6x - 6x\) = \(2x^2 + 8y^2 + 18 - 24y\)
ii. (3k – 4r – 2m)² – (3k + 4r – 2m)² = \([(3k)^2 + (- 4r)^2 + (- 2m)^2 + 2 (3k) (- 4r) + 2 (- 4r) (- 2m) + 2 (3k) (- 2m)] - [(3k)^2 + (4r)^2 + (- 2m)^2 + 2 (3k) (4r) + 2 (4r) (- 2m) + 2 (3k) (- 2m)]\) ... \([(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac]\) = \((9k^2 + 16r^2 + 4m^2 - 24kr + 16rm - 12km) - (9k^2 + 16r^2 + 4m^2 + 24kr - 16rm - 12km)\) = \(9k^2 + 16r^2 + 4m^2 - 24kr + 16rm - 12km - 9k^2 - 16r^2 - 4m^2 - 24kr + 16rm + 12km\) = \(9k^2 - 9k^2 + 16r^2 - 16r^2 + 4m^2 - 4m^2 - 24kr - 24kr + 16rm + 16rm - 12km + 12km\) = \(32rm - 48kr\)
iii. (7a – 6b + 5c)² + (7a + 6b – 5c)² = \([(7a)^2 + (- 6b)^2 + (5c)^2 + 2(7a) (-6b) + 2(-6b) (5c) + 2(7a) (5c)] + [(7a)^2 + (6b)^2 + (- 5c)^2 + 2 (7a) (6b) + 2 (6b) (- 5c) + 2 (7a) (- 5c)]\) ... \([(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac]\) = \(49a^2 + 36b^2 + 25c^2 - 84ab - 60bc + 70ac + 49a^2 + 36b^2 + 25c^2 + 84ab - 60bc - 70ac\) = \(49a^2 + 49a^2 + 36b^2 + 36b^2 + 25c^2 + 25c^2 - 84ab + 84ab - 60bc - 60bc + 70ac - 70ac\) = \(98a^2 + 72b^2 + 50c^2 - 120bc\) In simple words: This question involves simplifying expressions that combine or subtract expanded trinomials. It requires careful application of the \((a + b + c)^2\) formula and then combining like terms, paying close attention to the signs in subtraction problems.

🎯 Exam Tip: When simplifying expressions with subtraction, ensure that every term inside the subtracted bracket changes its sign. Double-check all calculations for combining like terms.

Maharashtra Board Class 8 Maths Chapter 5 Expansion Formulae Practice Set 5.4 Intext Questions And Activities

Question 1. Fill in the boxes with appropriate terms in the steps of expansion. (Textbook pg. no. 27)
(2p + 3m + 4n)²
= (2p)² + (3m)² + ______ + 2 × 2p x 3m + 2 × ______ × 4n + 2 × 2p x ______
= ______ + 9m² + ______ + 12pm + ______ + ______
Answer:
Solution:
(2p + 3m + 4n)²
= (2p)² + (3m)² + (4n)² + 2 x 2p x 3m + 2 x 3m x 4n + 2 x 2p x 4n
= \(4p^2 + 9m^2 + 16n^2 + 12pm + 24mn + 16pn\) In simple words: This question is an exercise in correctly applying the trinomial expansion formula \((a + b + c)^2\) by filling in the missing terms step-by-step, reinforcing the understanding of each component in the expansion.

🎯 Exam Tip: Practice filling in these types of blanks to ensure you understand each part of the expansion formula, especially the cross-product terms.