Algebraic Expressions and Identities — Questions and Answers
Welcome to HSLC Guru. This page gives the complete, step-by-step solutions to every exercise (Exercise 9.1 and Exercise 9.2) of Chapter 9, Algebraic Expressions and Identities, of ASSEB (Assam State School Education Board) Class 8 General Mathematics.
Summary
In this chapter we study the multiplication of algebraic expressions. The commutative, associative and distributive laws that hold for numbers hold for algebraic expressions too. When two or more monomials are multiplied, the numerical coefficient of the product equals the product of the numerical coefficients, and the algebraic coefficient equals the product of the algebraic coefficients, applying the law of indices $a^m \times a^n = a^{m+n}$.
To multiply a monomial by a binomial or trinomial, and a binomial by a binomial or trinomial, we use the distributive law $a(b + c) = ab + ac$, and then collect like terms.
A statement of equality that is true for every value of the variable is called an identity. The four identities used most often in algebra are:
$$(x + a)(x + b) = x^2 + (a + b)x + ab$$
$$(a + b)^2 = a^2 + 2ab + b^2$$
$$(a – b)^2 = a^2 – 2ab + b^2$$
$$(a + b)(a – b) = a^2 – b^2$$
Each of these can be verified geometrically. The figure below shows the identity $(a+b)^2 = a^2 + 2ab + b^2$ by dividing a square of side $(a + b)$ into four parts:
Summary: This ASSEB Class 8 General Mathematics Chapter 9 (Algebraic Expressions and Identities) solution covers multiplication of monomials, binomials and trinomials using the commutative, associative and distributive laws, and the four standard algebraic identities (x+a)(x+b)=x²+(a+b)x+ab, (a+b)²=a²+2ab+b², (a−b)²=a²−2ab+b² and (a+b)(a−b)=a²−b². Every question of Exercise 9.1 and Exercise 9.2 is solved step by step with KaTeX working and geometric figures.
Textbook Questions and Answers
Exercise 9.1
1. Find the product:
(i) $3x^2 \times 11xy \times \frac{2}{3}y^2$
Answer: Numerical coefficient $= 3 \times 11 \times \frac{2}{3} = 22$; variables $= x^{2+1}y^{1+2} = x^3y^3$. So the product is $22x^3y^3$.
(ii) $(-5x) \times 3a^2 \times (-3ax)$
Answer: Coefficient $= (-5)(3)(-3) = 45$; variables $= a^{2+1}x^{1+1} = a^3x^2$. Product $= 45a^3x^2$.
(iii) $(-3pq) \times (-15p^3q^3) \times q^2$
Answer: Coefficient $= (-3)(-15) = 45$; variables $= p^{1+3}q^{1+3+2} = p^4q^6$. Product $= 45p^4q^6$.
(iv) $3x(5x^2 + 8)$
Answer: $= 3x \times 5x^2 + 3x \times 8 = 15x^3 + 24x$.
(v) $\frac{2}{3}y(18y^2 – y)$
Answer: $= \frac{2}{3} \times 18\,y^3 – \frac{2}{3}y^2 = 12y^3 – \frac{2}{3}y^2$.
(vi) $(-8a^3)(a + 3b + 2c)$
Answer: $= -8a^4 – 24a^3b – 16a^3c$.
(vii) $(3mn – 2n)(-2m^2n)$
Answer: $= 3mn \times (-2m^2n) – 2n \times (-2m^2n) = -6m^3n^2 + 4m^2n^2$.
(viii) $(9x^2 + 4x + 3) \times 11x$
Answer: $= 99x^3 + 44x^2 + 33x$.
(ix) $(20a^2 – 3b^2 + ab) \times (-7b^2)$
Answer: $= -140a^2b^2 + 21b^4 – 7ab^3$.
(x) $3x^3y^2(xy + xy^3 – 2)$
Answer: $= 3x^4y^3 + 3x^4y^5 – 6x^3y^2$.
2. Find the product:
(i) $(x^2 + y)(3x^2y – y^2)$
Answer: $= 3x^4y – x^2y^2 + 3x^2y^2 – y^3 = 3x^4y + 2x^2y^2 – y^3$.
(ii) $(7x – 2y)(2x + 7y)$
Answer: $= 14x^2 + 49xy – 4xy – 14y^2 = 14x^2 + 45xy – 14y^2$.
(iii) $\left(\frac{1}{4}a^2 + 3b\right)\left(a^3 + \frac{2}{3}b^2\right)$
Answer: $= \frac{1}{4}a^5 + \frac{1}{6}a^2b^2 + 3a^3b + 2b^3$.
(iv) $(1.5x – 2.5y)(2.5x – 1.5y)$
Answer: $= 3.75x^2 – 2.25xy – 6.25xy + 3.75y^2 = 3.75x^2 – 8.5xy + 3.75y^2$.
(v) $(3x + 4y)(2x^2 + 3y + xy)$
Answer: $= 6x^3 + 9xy + 3x^2y + 8x^2y + 12y^2 + 4xy^2 = 6x^3 + 11x^2y + 4xy^2 + 9xy + 12y^2$.
(vi) $(2xy + 5x^2)(x^5y^4 – x^3y^2 + xy)$
Answer: $= 2x^6y^5 – 2x^4y^3 + 2x^2y^2 + 5x^7y^4 – 5x^5y^2 + 5x^3y$.
(vii) $(3a^2b^2 – 4c)(a^3b^3 + 2a^4b^3c^3 – 6abc)$
Answer: $= 3a^5b^5 + 6a^6b^5c^3 – 18a^3b^3c – 4a^3b^3c – 8a^4b^3c^4 + 24abc^2 = 6a^6b^5c^3 + 3a^5b^5 – 8a^4b^3c^4 – 22a^3b^3c + 24abc^2$.
(viii) $(4x^2y – 5xy^2 + 3xy)(3x^3y – 2)$
Answer: $= 12x^5y^2 – 8x^2y – 15x^4y^3 + 10xy^2 + 9x^4y^2 – 6xy = 12x^5y^2 – 15x^4y^3 + 9x^4y^2 – 8x^2y + 10xy^2 – 6xy$.
(ix) $(2x + 3y + z)(5x + 2y + 1)$
Answer: $= 10x^2 + 4xy + 2x + 15xy + 6y^2 + 3y + 5xz + 2yz + z = 10x^2 + 19xy + 6y^2 + 5xz + 2yz + 2x + 3y + z$.
(x) $(3x – 2y + z)(3x + 2y – z)$
Answer: $(3x – 2y + z)(3x + 2y – z) = \{3x – (2y – z)\}\{3x + (2y – z)\} = (3x)^2 – (2y – z)^2 = 9x^2 – 4y^2 + 4yz – z^2$.
3. Simplify the following expressions:
(i) $3x(5x + 8) – 10x$
Answer: $= 15x^2 + 24x – 10x = 15x^2 + 14x$.
(ii) $(2m + 3m^2)(-2mn)$
Answer: $= -4m^2n – 6m^3n$.
(iii) $8(3a + 4b) + 5$
Answer: $= 24a + 32b + 5$.
(iv) $2x^2(4x – 1) + 3x(x – 3)$
Answer: $= 8x^3 – 2x^2 + 3x^2 – 9x = 8x^3 + x^2 – 9x$.
4. Simplify:
(i) $(p + q^2)(q^2 – p) + 15$
Answer: $(p + q^2)(q^2 – p) = (q^2 + p)(q^2 – p) = q^4 – p^2$. So the expression $= q^4 – p^2 + 15$.
(ii) $(a – b)(a^2 + ab + b^2) + 3b^3$
Answer: $(a – b)(a^2 + ab + b^2) = a^3 – b^3$. So $= a^3 – b^3 + 3b^3 = a^3 + 2b^3$.
(iii) $y^2(y^3 + 3x) + y(2xy + y^2)$
Answer: $= y^5 + 3xy^2 + 2xy^2 + y^3 = y^5 + y^3 + 5xy^2$.
(iv) $\left(\frac{2}{3}x^4y^3 + \frac{4}{9}xy^3\right) \times \frac{1}{4} – \frac{1}{6}x^4y^3$
Answer: $= \frac{2}{3} \cdot \frac{1}{4}x^4y^3 + \frac{4}{9} \cdot \frac{1}{4}xy^3 – \frac{1}{6}x^4y^3 = \frac{1}{6}x^4y^3 + \frac{1}{9}xy^3 – \frac{1}{6}x^4y^3 = \frac{1}{9}xy^3$.
(v) $y^3(4y + 5) – (2y + 1)(y^3 + 2y^2 + 1)$
Answer: $y^3(4y + 5) = 4y^4 + 5y^3$ and $(2y + 1)(y^3 + 2y^2 + 1) = 2y^4 + 5y^3 + 2y^2 + 2y + 1$. Subtracting, $= 4y^4 + 5y^3 – 2y^4 – 5y^3 – 2y^2 – 2y – 1 = 2y^4 – 2y^2 – 2y – 1$.
(vi) $(1.2l – 2.5m)(2.5l + 0.2m + 1.2) + 0.06l + 7m$
Answer: $(1.2l – 2.5m)(2.5l + 0.2m + 1.2) = 3l^2 + 0.24lm + 1.44l – 6.25lm – 0.5m^2 – 3m$. Adding $0.06l + 7m$, $= 3l^2 – 6.01lm – 0.5m^2 + 1.5l + 4m$.
Activity — Completing the multiplication table
Question: Fill up the blanks of the following multiplication table.
Answer: Putting (row expression) × (column expression) in each cell, the completed table is:
| × | 6x | –3y | 7xy | –5x²y | 8x²y³ |
|---|---|---|---|---|---|
| 6x | 36x² | –18xy | 42x²y | –30x³y | 48x³y³ |
| –3y | –18xy | 9y² | –21xy² | 15x²y² | –24x²y⁴ |
| 7xy | 42x²y | –21xy² | 49x²y² | –35x³y² | 56x³y⁴ |
| –5x²y | –30x³y | 15x²y² | –35x³y² | 25x⁴y² | –40x⁴y⁴ |
| 8x²y³ | 48x³y³ | –24x²y⁴ | 56x³y⁴ | –40x⁴y⁴ | 64x⁴y⁶ |
Exercise 9.2
The figure below shows the identity $(x + a)(x + b) = x^2 + (a + b)x + ab$ by dividing a rectangle into four parts:
1. Find the product using the identity $(x + a)(x + b) = x^2 + (a + b)x + ab$:
(i) $(x + 7)(x + 5)$
Answer: $= x^2 + (7 + 5)x + 7 \times 5 = x^2 + 12x + 35$.
(ii) $(7x + 2y)(7x + 6y)$
Answer: $= (7x)^2 + (2y + 6y)(7x) + 2y \times 6y = 49x^2 + 56xy + 12y^2$.
(iii) $(4x^3 + 8)(4x^3 + 10)$
Answer: $= (4x^3)^2 + (8 + 10)(4x^3) + 8 \times 10 = 16x^6 + 72x^3 + 80$.
(iv) $(4k^2 – 3k)(4k^2 – 7k)$
Answer: $= (4k^2)^2 + (-3k – 7k)(4k^2) + (-3k)(-7k) = 16k^4 – 40k^3 + 21k^2$.
(v) $\left(\frac{a}{2} + \frac{1}{2}\right)\left(\frac{a}{2} – \frac{1}{4}\right)$
Answer: $= \left(\frac{a}{2}\right)^2 + \left(\frac{1}{2} – \frac{1}{4}\right)\frac{a}{2} + \frac{1}{2} \times \left(-\frac{1}{4}\right) = \frac{a^2}{4} + \frac{a}{8} – \frac{1}{8}$.
(vi) $\left(\frac{n^2}{5} – 0.6\right)\left(\frac{n^2}{5} + 1.6\right)$
Answer: $= \left(\frac{n^2}{5}\right)^2 + (-0.6 + 1.6)\frac{n^2}{5} + (-0.6)(1.6) = \frac{n^4}{25} + \frac{n^2}{5} – 0.96$.
(vii) $98 \times 97$
Answer: $98 \times 97 = (100 – 2)(100 – 3) = 100^2 + (-2 – 3)(100) + (-2)(-3) = 10000 – 500 + 6 = 9506$.
(viii) $501 \times 503$
Answer: $501 \times 503 = (500 + 1)(500 + 3) = 500^2 + (1 + 3)(500) + 1 \times 3 = 250000 + 2000 + 3 = 252003$.
2. Evaluate using the identity $(a + b)^2 = a^2 + 2ab + b^2$:
(i) $(x + 5)^2$
Answer: $= x^2 + 2 \times x \times 5 + 5^2 = x^2 + 10x + 25$.
(ii) $(5x + 4y)^2$
Answer: $= (5x)^2 + 2 \times 5x \times 4y + (4y)^2 = 25x^2 + 40xy + 16y^2$.
(iii) $(3a^3 + 4a^2)^2$
Answer: $= (3a^3)^2 + 2 \times 3a^3 \times 4a^2 + (4a^2)^2 = 9a^6 + 24a^5 + 16a^4$.
(iv) $\left(3x + \frac{1}{3x}\right)^2$
Answer: $= (3x)^2 + 2 \times 3x \times \frac{1}{3x} + \left(\frac{1}{3x}\right)^2 = 9x^2 + 2 + \frac{1}{9x^2}$.
(v) $\left(\frac{p}{q} + \frac{q}{p}\right)^2$
Answer: $= \frac{p^2}{q^2} + 2 \times \frac{p}{q} \times \frac{q}{p} + \frac{q^2}{p^2} = \frac{p^2}{q^2} + 2 + \frac{q^2}{p^2}$.
(vi) $502^2$
Answer: $502^2 = (500 + 2)^2 = 500^2 + 2 \times 500 \times 2 + 2^2 = 250000 + 2000 + 4 = 252004$.
(vii) $(9.5)^2$
Answer: $(9.5)^2 = (9 + 0.5)^2 = 81 + 2 \times 9 \times 0.5 + 0.25 = 81 + 9 + 0.25 = 90.25$.
(viii) $\left(4\frac{1}{8}\right)^2$
Answer: $\left(4 + \frac{1}{8}\right)^2 = 4^2 + 2 \times 4 \times \frac{1}{8} + \left(\frac{1}{8}\right)^2 = 16 + 1 + \frac{1}{64} = 17\frac{1}{64} = \frac{1089}{64}$.
3. Evaluate using the identity $(a – b)^2 = a^2 – 2ab + b^2$:
(i) $(x – 7)^2$
Answer: $= x^2 – 2 \times x \times 7 + 7^2 = x^2 – 14x + 49$.
(ii) $(6x – 5)^2$
Answer: $= (6x)^2 – 2 \times 6x \times 5 + 5^2 = 36x^2 – 60x + 25$.
(iii) $(10x^2 – 3y)^2$
Answer: $= (10x^2)^2 – 2 \times 10x^2 \times 3y + (3y)^2 = 100x^4 – 60x^2y + 9y^2$.
(iv) $(p^2 – q^2)^2$
Answer: $= (p^2)^2 – 2 \times p^2 \times q^2 + (q^2)^2 = p^4 – 2p^2q^2 + q^4$.
(v) $(a^2x – ax^2)^2$
Answer: $= (a^2x)^2 – 2 \times a^2x \times ax^2 + (ax^2)^2 = a^4x^2 – 2a^3x^3 + a^2x^4$.
(vi) $\left(x^2 – \frac{1}{x^2}\right)^2$
Answer: $= (x^2)^2 – 2 \times x^2 \times \frac{1}{x^2} + \left(\frac{1}{x^2}\right)^2 = x^4 – 2 + \frac{1}{x^4}$.
(vii) $296^2$
Answer: $296^2 = (300 – 4)^2 = 90000 – 2400 + 16 = 87616$.
(viii) $1999^2$
Answer: $1999^2 = (2000 – 1)^2 = 4000000 – 4000 + 1 = 3996001$.
4. Find the product using the identity $(a + b)(a – b) = a^2 – b^2$:
(i) $(y + 11)(y – 11)$
Answer: $= y^2 – 11^2 = y^2 – 121$.
(ii) $(2x + 3)(2x – 3)$
Answer: $= (2x)^2 – 3^2 = 4x^2 – 9$.
(iii) $(6 + m^2)(m^2 – 6)$
Answer: $= (m^2 + 6)(m^2 – 6) = (m^2)^2 – 6^2 = m^4 – 36$.
(iv) $(ax^2 – by)(ax^2 + by)$
Answer: $= (ax^2)^2 – (by)^2 = a^2x^4 – b^2y^2$.
(v) $(1 – x^m)(1 + x^m)$
Answer: $= 1^2 – (x^m)^2 = 1 – x^{2m}$.
(vi) $61 \times 59$
Answer: $61 \times 59 = (60 + 1)(60 – 1) = 60^2 – 1^2 = 3600 – 1 = 3599$.
(vii) $106 \times 94$
Answer: $106 \times 94 = (100 + 6)(100 – 6) = 10000 – 36 = 9964$.
(viii) $9.5 \times 8.5$
Answer: $9.5 \times 8.5 = (9 + 0.5)(9 – 0.5) = 81 – 0.25 = 80.75$.
5. Find the product using suitable identities:
(i) $(3x – 5m)(3x – 5m)$
Answer: $= (3x – 5m)^2 = (3x)^2 – 2 \times 3x \times 5m + (5m)^2 = 9x^2 – 30xm + 25m^2$.
(ii) $(4m + 3)(4m + 2)$
Answer: $= (4m)^2 + (3 + 2)(4m) + 3 \times 2 = 16m^2 + 20m + 6$.
(iii) $(9 + 4n)^2$
Answer: $= 9^2 + 2 \times 9 \times 4n + (4n)^2 = 81 + 72n + 16n^2$.
(iv) $\left(6x + \frac{1}{3}\right)(6x + 3)$
Answer: $= (6x)^2 + \left(\frac{1}{3} + 3\right)(6x) + \frac{1}{3} \times 3 = 36x^2 + \frac{10}{3} \times 6x + 1 = 36x^2 + 20x + 1$.
(v) $(4ab – c)(4ab + c)$
Answer: $= (4ab)^2 – c^2 = 16a^2b^2 – c^2$.
(vi) $\left(x – \frac{x}{2}\right)^2$
Answer: $x – \frac{x}{2} = \frac{x}{2}$, so $\left(\frac{x}{2}\right)^2 = \frac{x^2}{4}$.
(vii) $\left(\frac{a^2}{2} + \frac{b^2}{4}\right)\left(\frac{a^2}{2} + \frac{b^2}{4}\right)$
Answer: $= \left(\frac{a^2}{2} + \frac{b^2}{4}\right)^2 = \left(\frac{a^2}{2}\right)^2 + 2 \times \frac{a^2}{2} \times \frac{b^2}{4} + \left(\frac{b^2}{4}\right)^2 = \frac{a^4}{4} + \frac{a^2b^2}{4} + \frac{b^4}{16}$.
(viii) $(0.5x^2 – 0.2y^2)^2$
Answer: $= (0.5x^2)^2 – 2 \times 0.5x^2 \times 0.2y^2 + (0.2y^2)^2 = 0.25x^4 – 0.2x^2y^2 + 0.04y^4$.
(ix) $(-9x^2 + y^3)(9x^2 + y^3)$
Answer: $= (y^3 – 9x^2)(y^3 + 9x^2) = (y^3)^2 – (9x^2)^2 = y^6 – 81x^4$.
(x) $\left(\frac{y^2}{2} – 4\right)\left(\frac{y^2}{2} + 6\right)$
Answer: $= \left(\frac{y^2}{2}\right)^2 + (-4 + 6)\frac{y^2}{2} + (-4)(6) = \frac{y^4}{4} + y^2 – 24$.
(xi) $\left(7x^2 + \frac{1}{3}\right)^2$
Answer: $= (7x^2)^2 + 2 \times 7x^2 \times \frac{1}{3} + \left(\frac{1}{3}\right)^2 = 49x^4 + \frac{14}{3}x^2 + \frac{1}{9}$.
(xii) $(x + y + z)(x + y – z)$
Answer: $= \{(x + y) + z\}\{(x + y) – z\} = (x + y)^2 – z^2 = x^2 + 2xy + y^2 – z^2$.
(xiii) $1002 \times 999$
Answer: $1002 \times 999 = (1000 + 2)(1000 – 1) = 1000^2 + (2 – 1)(1000) + 2 \times (-1) = 1000000 + 1000 – 2 = 1000998$.
(xiv) $(10.2)^2$
Answer: $(10.2)^2 = (10 + 0.2)^2 = 100 + 4 + 0.04 = 104.04$.
(xv) $79^2$
Answer: $79^2 = (80 – 1)^2 = 6400 – 160 + 1 = 6241$.
(xvi) $6.2 \times 5.8$
Answer: $6.2 \times 5.8 = (6 + 0.2)(6 – 0.2) = 36 – 0.04 = 35.96$.
6. Simplify:
(i) $\left(x + \frac{1}{x}\right)\left(2x + \frac{1}{x}\right)$
Answer: $= 2x^2 + 1 + 2 + \frac{1}{x^2} = 2x^2 + 3 + \frac{1}{x^2}$.
(ii) $(2l + m)^2 – (2l – m)^2$
Answer: Using $(A + B)^2 – (A – B)^2 = 4AB$, $= 4 \times 2l \times m = 8lm$.
(iii) $(a^2b + ab^2)^2 – 6a^3b^3$
Answer: $(a^2b + ab^2)^2 = a^4b^2 + 2a^3b^3 + a^2b^4$. So $= a^4b^2 + 2a^3b^3 + a^2b^4 – 6a^3b^3 = a^4b^2 – 4a^3b^3 + a^2b^4$.
(iv) $(x + y)(x – y) + (y + z)(y – z) + (z + x)(z – x)$
Answer: $= (x^2 – y^2) + (y^2 – z^2) + (z^2 – x^2) = 0$.
(v) $(5a – 6b)^2 + 20ab – (6b + 5a)^2$
Answer: $(5a – 6b)^2 – (5a + 6b)^2 = \{(5a – 6b) + (5a + 6b)\}\{(5a – 6b) – (5a + 6b)\} = (10a)(-12b) = -120ab$. So $= -120ab + 20ab = -100ab$.
(vi) $(4p^2 + 5q^2)(4p^2 – 5q^2) + (2p^2 – 5q^2)^2$
Answer: $= (16p^4 – 25q^4) + (4p^4 – 20p^2q^2 + 25q^4) = 20p^4 – 20p^2q^2$.
(vii) $(2x – 5)(2x + 3) – (x – 2)^2 + 29$
Answer: $(2x – 5)(2x + 3) = 4x^2 – 4x – 15$ and $(x – 2)^2 = x^2 – 4x + 4$. So $= 4x^2 – 4x – 15 – x^2 + 4x – 4 + 29 = 3x^2 + 10$.
(viii) $\left(\frac{x}{3} – \frac{3y}{4}\right)\left(\frac{x}{3} + \frac{3y}{4}\right) + \left(\frac{3y}{4} + 3x\right)\left(\frac{3y}{4} + x\right)$
Answer: First part $= \frac{x^2}{9} – \frac{9y^2}{16}$; second part $= \frac{9y^2}{16} + (3x + x)\frac{3y}{4} + 3x^2 = \frac{9y^2}{16} + 3xy + 3x^2$. Adding, $= \frac{x^2}{9} + 3x^2 + 3xy = \frac{28x^2}{9} + 3xy$.
(ix) $\left(\frac{x}{5} + \frac{y}{5}\right)^2 + 2\left(\frac{x}{5} + \frac{y}{5}\right)\left(\frac{x}{5} – \frac{y}{5}\right) + \left(\frac{x}{5} – \frac{y}{5}\right)^2$
Answer: This is of the form $A^2 + 2AB + B^2 = (A + B)^2$ with $A = \frac{x}{5} + \frac{y}{5}$ and $B = \frac{x}{5} – \frac{y}{5}$. Since $A + B = \frac{2x}{5}$, the value $= \left(\frac{2x}{5}\right)^2 = \frac{4x^2}{25}$.
(x) $2.89 \times 2.89 + 0.22 \times 2.89 + 0.0121$
Answer: Here $a = 2.89$, $b = 0.11$ (since $0.22 = 2 \times 0.11$ and $0.0121 = 0.11^2$). So $= a^2 + 2ab + b^2 = (2.89 + 0.11)^2 = 3^2 = 9$.
(xi) $\dfrac{0.25 – 2 \times 0.5 \times 3.5 + 12.25}{3}$
Answer: Numerator $= (0.5)^2 – 2 \times 0.5 \times 3.5 + (3.5)^2 = (0.5 – 3.5)^2 = 9$. So $= \frac{9}{3} = 3$.
(xii) $\dfrac{4.68 \times 4.68 – 3.32 \times 3.32}{1.36}$
Answer: Numerator $= (4.68)^2 – (3.32)^2 = (4.68 + 3.32)(4.68 – 3.32) = 8 \times 1.36 = 10.88$. So $= \frac{10.88}{1.36} = 8$.
7. Show that:
(i) $(a – b + c – d)^2 – (a + b – c + d)^2 + 4a(b + d) = 4ca$
Answer: Let $P = a – b + c – d$ and $Q = a + b – c + d$. Then $P + Q = 2a$ and $P – Q = 2(c – b – d)$. So $P^2 – Q^2 = (P + Q)(P – Q) = 2a \times 2(c – b – d) = 4ac – 4ab – 4ad$. Adding $4a(b + d) = 4ab + 4ad$, LHS $= 4ac – 4ab – 4ad + 4ab + 4ad = 4ac = 4ca =$ RHS. (Proved)
(ii) $(1.5x^2 + 1.2y)^2 – (1.5x^2 – 1.2y)^2 = 7.2x^2y$
Answer: Using $(A + B)^2 – (A – B)^2 = 4AB$ with $A = 1.5x^2$, $B = 1.2y$: LHS $= 4 \times 1.5x^2 \times 1.2y = 7.2x^2y =$ RHS. (Proved)
(iii) $\left(\frac{2}{3}x^2 + 5\right)^2 – 25 = \frac{4}{9}x^4 + \frac{20}{3}x^2$
Answer: LHS $= \left(\frac{2}{3}x^2\right)^2 + 2 \times \frac{2}{3}x^2 \times 5 + 25 – 25 = \frac{4}{9}x^4 + \frac{20}{3}x^2 =$ RHS. (Proved)
(iv) $(a^2 + b^2)(a^2 – b^2) + (b^2 + c^2)(b^2 – c^2) + 2c^2(c^2 – a^2) = (c^2 – a^2)^2$
Answer: LHS $= (a^4 – b^4) + (b^4 – c^4) + 2c^4 – 2a^2c^2 = a^4 – c^4 + 2c^4 – 2a^2c^2 = a^4 + c^4 – 2a^2c^2 = (c^2 – a^2)^2 =$ RHS. (Proved)
8. Solve the following problems using suitable identities:
(i) Find the area of a rectangular playground of length $(x + 8)$ metres and breadth 3 metres less than the length.
Answer: Breadth $= (x + 8) – 3 = (x + 5)$ m. Area $= (x + 8)(x + 5) = x^2 + (8 + 5)x + 40 = x^2 + 13x + 40$ square metres.
(ii) Find the area of a square garden whose length is $\left(2x + \frac{1}{4}\right)$ metres.
Answer: Area $= \left(2x + \frac{1}{4}\right)^2 = (2x)^2 + 2 \times 2x \times \frac{1}{4} + \left(\frac{1}{4}\right)^2 = 4x^2 + x + \frac{1}{16}$ square metres.
(iii) A cultivator grows Joha and Bora paddy on two square-shaped plots. The side of the plot where Joha is grown is 5 m more than the side of the plot where Bora is grown. Find the difference between the areas of the two plots.
Answer: Let the side of the Bora plot be $x$ m, so the side of the Joha plot is $(x + 5)$ m. Difference of areas $= (x + 5)^2 – x^2 = \{(x + 5) + x\}\{(x + 5) – x\} = (2x + 5)(5) = (10x + 25)$ square metres.
(iv) The cost of painting a wall is Rs. 9.00 per square metre. Find the cost of painting a wall of length 107 m and breadth 93 m.
Answer: Area of the wall $= 107 \times 93 = (100 + 7)(100 – 7) = 10000 – 49 = 9951$ square metres. Cost $= 9951 \times 9 = $ Rs. $89559$.
(v) Find the area of a square field whose length is 197 m.
Answer: Area $= 197^2 = (200 – 3)^2 = 40000 – 1200 + 9 = 38809$ square metres.
(vi) If $x + \frac{1}{x} = 3$, find the values of $x^2 + \frac{1}{x^2}$ and $x^4 + \frac{1}{x^4}$.
Answer: $x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 – 2 = 9 – 2 = 7$. Then $x^4 + \frac{1}{x^4} = \left(x^2 + \frac{1}{x^2}\right)^2 – 2 = 49 – 2 = 47$.
(vii) If $2x – \frac{1}{2x} = 2$, find the values of $4x^2 + \frac{1}{4x^2}$ and $16x^4 – \frac{1}{16x^4}$.
Answer: $4x^2 + \frac{1}{4x^2} = \left(2x – \frac{1}{2x}\right)^2 + 2 = 4 + 2 = 6$. Also $\left(2x + \frac{1}{2x}\right)^2 = \left(2x – \frac{1}{2x}\right)^2 + 4 = 8$, so $2x + \frac{1}{2x} = 2\sqrt{2}$; hence $4x^2 – \frac{1}{4x^2} = \left(2x – \frac{1}{2x}\right)\left(2x + \frac{1}{2x}\right) = 2 \times 2\sqrt{2} = 4\sqrt{2}$. Therefore $16x^4 – \frac{1}{16x^4} = \left(4x^2 – \frac{1}{4x^2}\right)\left(4x^2 + \frac{1}{4x^2}\right) = 4\sqrt{2} \times 6 = 24\sqrt{2}$.
(viii) If $a – b = 10$ and $ab = 11$, find the value of $a + b$.
Answer: $(a + b)^2 = (a – b)^2 + 4ab = 10^2 + 4 \times 11 = 100 + 44 = 144$. So $a + b = \sqrt{144} = 12$ (positive value).
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. $(a + b)^2$ is equal to—
(a) $a^2 + b^2$ (b) $a^2 + 2ab + b^2$ (c) $a^2 – 2ab + b^2$ (d) $a^2 – b^2$
Answer: (b) $a^2 + 2ab + b^2$.
2. $(a + b)(a – b)$ is equal to—
(a) $a^2 + b^2$ (b) $2ab$ (c) $a^2 – b^2$ (d) $a^2 – 2ab + b^2$
Answer: (c) $a^2 – b^2$.
3. The product $5x \times (-8x^3y)$ is—
(a) $-40x^4y$ (b) $40x^4y$ (c) $-40x^3y$ (d) $-13x^4y$
Answer: (a) $-40x^4y$.
4. $(x + 3)(x + 4)$ is equal to—
(a) $x^2 + 12x + 7$ (b) $x^2 + 7x + 12$ (c) $x^2 + 7x + 7$ (d) $x^2 + 12$
Answer: (b) $x^2 + 7x + 12$.
5. The value of $102^2$ is—
(a) $10404$ (b) $10204$ (c) $10440$ (d) $10004$
Answer: (a) $10404$ (since $(100 + 2)^2 = 10000 + 400 + 4$).
6. The value of $99 \times 101$ is—
(a) $9999$ (b) $10000$ (c) $9899$ (d) $10099$
Answer: (a) $9999$ (since $(100 – 1)(100 + 1) = 10000 – 1$).
7. The value of $(2x)^2$ is—
(a) $2x^2$ (b) $4x^2$ (c) $4x$ (d) $2x$
Answer: (b) $4x^2$.
8. An algebraic expression having three terms is called a—
(a) monomial (b) binomial (c) trinomial (d) polynomial
Answer: (c) trinomial.
9. $a^m \times a^n$ equals—
(a) $a^{m-n}$ (b) $a^{mn}$ (c) $a^{m+n}$ (d) $a^{m/n}$
Answer: (c) $a^{m+n}$.
10. $(a – b)^2 + 2ab$ equals—
(a) $a^2 + b^2$ (b) $a^2 – b^2$ (c) $2ab$ (d) $a^2 + 2ab + b^2$
Answer: (a) $a^2 + b^2$ (since $a^2 – 2ab + b^2 + 2ab = a^2 + b^2$).
Fill in the Blanks
1. $(a + b)^2 – (a – b)^2 = $ ______.
Answer: $4ab$.
2. $(a + b)^2 + (a – b)^2 = $ ______.
Answer: $2(a^2 + b^2)$.
3. $(x + a)(x + b) = x^2 + $ ______ $x + ab$.
Answer: $(a + b)$.
4. An algebraic expression having one term is called a ______.
Answer: monomial.
5. $101 \times 99 = $ ______.
Answer: $9999$.
True or False
1. $(a + b)^2 = a^2 + b^2$.
Answer: False. ($(a + b)^2 = a^2 + 2ab + b^2$)
2. Unlike terms can be added or subtracted.
Answer: False. Only like terms can be added or subtracted.
3. $(a + b)(a – b) = a^2 – b^2$.
Answer: True.
4. $x \times x = x^2$.
Answer: True.
5. An identity is true for every value of the variable.
Answer: True.
Short Answer Questions
1. Find the product $4x^2 \times 3xy$.
Answer: $= 12x^3y$.
2. Simplify $(x + 2)^2 – (x – 2)^2$.
Answer: Using $(A + B)^2 – (A – B)^2 = 4AB$, $= 4 \times x \times 2 = 8x$.
3. Evaluate $103^2$ using an identity.
Answer: $103^2 = (100 + 3)^2 = 10000 + 600 + 9 = 10609$.
4. If $a + b = 7$ and $ab = 12$, find $a^2 + b^2$.
Answer: $a^2 + b^2 = (a + b)^2 – 2ab = 49 – 24 = 25$.
Key Terms
| Term | Meaning |
|---|---|
| Algebraic expression | A mathematical expression formed of variables and constants |
| Identity | An equality true for every value of the variable |
| Monomial | An expression with one term |
| Binomial | An expression with two terms |
| Trinomial | An expression with three terms |
| Coefficient | The numerical or algebraic factor of a term |
| Distributive law | $a(b + c) = ab + ac$ |
| Like terms | Terms with the same variables and same exponents |
| Product | The result of multiplying two or more expressions |
| Square of a binomial | $(a + b)^2 = a^2 + 2ab + b^2$ |