Class 8 Algebraic Expressions and Identities Exercise 6.7

\begin{array}{l} \text {Q1. Find the following products: } \\ \text {(i) } \quad (x+4)(x+7) \\ \text {(ii) } \quad (x-11)(x+4) \\ \text {(iii) } \quad (x+7)(x-5) \\ \text {(iv) } \quad (x-3)(x-2) \\ \text {(v) } \quad (y^{2}-4)(y^{2}-3) \\ \text {(vi) } \quad (x+4 / 3)(x+3 / 4) \\ \text {(vii) } \quad (3 x+5)(3 x+11) \\ \text {(viii) } \quad (2 x^{2}-3)(2 x^{2}+5) \\ \text {(ix) } \quad (z^{2}+2)(z^{2}-3) \\ \text {(x) } \quad (3 x-4 y)(2 x-4 y) \\ \text {(xi) } \quad (3 x^{2}-4 x y)(3 x^{2}-3 x y) \\ \text {(xii) } \quad (x+1 / 5)(x+5) \\ \text {(xiii) } \quad (\mathbf{z}+3 / 4)(\mathbf{z}+4 / 3) \\ \text {(xiv) } \quad (x^{2}+4)(x^{2}+9) \\ \text {(xy) } \quad (y^{2}+12)(y^{2}+6) \\ \text {(xvi) } \quad (y^{2}+5 / 7)(y^{2}-14 / 5) \\ \text {(xvii) } \quad (p^{2}+16)(p^{2}-1 / 4) \\ \end{array}\begin{array}{l} \text {Sol. } \\ \text {(i) Given expression is } (x+4)(x+7) \\ =(x^{2}+(4+7) x+4 \times 7) \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b] \\ =x^{2}+11 x+28 \\ \end{array}\begin{array}{l} \text {(ii) Given expression is } (x-11)(x+4) \\ =x^{2}+(4-11) x-11 \times 4 \quad \quad [\text {Using: } (x-a)(x+b)=x^{2}+(b-a) x-a b ] \\ =x^{2}-7x-44 \\ \end{array}\begin{array}{l} \text {(iii) Given expression is} (x+7)(x-5) \\ =(x^{2}+(7-5) x-7 \times 5) \quad \quad [\text {Using: } (x+a)(x-b)=x^{2}+(a-b) x-a b] \\ =x^{2}+2 x-35 \\ \end{array}\begin{array}{l} \text {(iv) Given expression is } (x-3)(x-2) \\=(x^{2}-(3+2) x+3 \times 2) \quad \quad [\text {Using: } (x-a)(x-b)=x^{2}-(a+b) x+a b] \\ =x^{2}-5x+6 \\ \end{array}\begin{array}{l} \text {(v) Given expression is } (y^{2}-4)(y^{2}-3) \\=(y^{2})^{2}-(4+3)(y^{2})+4 \times 3) \quad \quad [\text {Using: } (x-a)(x-b)=x^{2}-(a+b) x-a b] \\ =y^{4}-7 y^{2}+12 \\ \end{array}\begin{array}{l} \text {(vi) Given expression is } (x+\frac{4}{3})(x+\frac{3}{4}) \\ =x^{2}+(\frac{4}{3}+\frac{3}{4}) x+\frac{4}{3} \times \frac{3}{4} \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b] \\ =x^{2}+\frac{25}{12} x+1\\ \end{array}\begin{array}{l} \text {(vii) Given expression is } (3 x+5)(3 x+11) \\ =(3 x)^{2}+(5+11)(3 x)+5 \times 11 \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b] \\ =9 x^{2}+48 x+55 \\ \end{array}\begin{array}{l} \text {(viii) Given expression is } (2 x^{2}-3)(2 x^{2}+5) \\ =(2 x^{2})^{2}+(5-3)(2 x^{2})-3 \times 5 \quad \quad [\text {Using: } (x-a)(x+b)=x^{2}+(b-a) x-a b] \\ =4 x^{4}+4 x^{2}-15 \\ \end{array}\begin{array}{l} \text { (ix) Given expression is } (z^{2}+2)(z^{2}-3) \\ =(z^{2})^{2}+(2-3)(z^{2})-2 \times 3 \quad \quad [\text {Using: } (x+a)(x-b)=x^{2}+(a-b) x-a b] \\ =z^{4}-z^{2}-6 \\ \end{array}\begin{array}{l} \text {(x) Given expression is } \quad (3 x-4 y)(2 x-4 y) \\=(-)(4 y-3 x)(-1)(4y-2x) \quad \quad [\text {Taking (-1) common from both parentheses} ] \\=(4y – 3x)(4y – 2x) \\=(4 y)^{2}-(3 x+2 x)(4 y)+3 x \times 2 x \quad \quad [\text {Using: } (x-a)(x-b)=x^{2}-(a+b) x-a b] \\=16 y^{2}-(12 x y+8 x y)+6 x^{2} \\ =16 y^{2}-20 x y+6 x^{2} \\ \end{array}\begin{array}{l} \text {(xi) Given expression is } \quad (3 x^{2}-4 x y)(3 x^{2}-3 x y) \\=(3 x^{2})^{2}-(4 x y+3 x y)(3 x^{2})+4 x y \times 3 x y \quad \quad [\text {Using: } (x-a)(x-b)=x^{2}-(a+b) x-a b] \\=9 x^{4}-(12 x^{3} y+9 x^{3} y)+12 x^{2} y^{2} \\=9 x^{4}-21x^{3}y+12x^{2}y^{2} \\ \end{array}\begin{array}{l}\text {(xii) Given expression is } \quad (x+\frac{1}{5})(x+5) \\=(x^{2}+(\frac{1}{5}+5) x+\frac{1}{5} \times 5) \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b ] \\=x^{2}+\frac{26}{5} x+1 \\ \end{array}\begin{array}{l} \text {(xiii) Given expression is } \quad (z+\frac{3}{4})(z+\frac{4}{3}) \\=z^{2}+(\frac{3}{4}+\frac{4}{3}) x+\frac{3}{4} \times \frac{4}{3} \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b ] \\=z^{2}+\frac{25}{12} x+1\\ \end{array}\begin{array}{l} \text {(xiv) Given expression is } \quad (x^{2}+4)(x^{2}+9) \\ =(x^{2})^{2}+(4+9)(x^{2})+4 \times 9 \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b ] \\ =x^{4}+13 x^{2}+36\\ \end{array}\begin{array}{l} \text {(xv) Given expression is } \quad (y^{2}+12)(y^{2}+6) \\=(y^{2})^{2}+(12+6)(y^{2})+12 \times 6 \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b ] \\=y^{4}+18 x y^{2}+72 \\ \end{array}\begin{array}{l} \text {(xvi) Given expression is } \quad (y^{2}+\frac{5}{7})(y^{2}-\frac{14}{5}) \\=(y^{2})^{2}+(\frac{5}{7}-\frac{14}{5})(y^{2})-\frac{5}{7} \times \frac{14}{5} \quad \quad [\text {Using: } (x+a)(x-b)=x^{2}+(a-b) x-a b ] \\=y^{4}-\frac{73}{35}y^{2}-2 \\ \end{array}\begin{array}{l} \text {(xvii) Given expression is } \quad (p^{2}+16)(p^{2}-\frac{1}{4}) \\=(p^{2})^{2}+(16-\frac{1}{4})(p^{2})-16 \times \frac{1}{4} \quad \quad [\text {Using: } (x+a)(x-b)=x^{2}+(a-b) x-a b ] \\=p^{4}+\frac{63}{4} p^{2}-4 \\ \end{array}\begin{array}{l} \text {Q2. Evaluate the following: } \\ \text {(i) } \quad 102 \times 106 \\ \text {(ii) } \quad 109 \times 107 \\ \text {(iii) } \quad 35 \times 37 \\ \text {(iv) } \quad 53 \times 55 \\ \text {(v) } \quad 103 \times 96 \\ \text {(vi) } \quad 34 \times 36 \\ \text {(vii) } \quad 994 \times 1006 \\ \end{array}\begin{array}{l} \text {Sol. } \\ \text {(i) Given that } \quad 102 \times 106 \\ =(100+2)(100+6) \\ =100^{2}+(2+6) 100+2 \times 6 \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b ] \\ =10000+800+12=10812 \\ \end{array}\begin{array}{l} \text {(ii) Given that } \quad 109 \times 107 \\ =(100+9)(100+7) \\ =100^{2}+(9+7) 100+9 \times 7 \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b ] \\ =10000+1600+63=11663\\ \end{array}\begin{array}{l} \text {(iii) Given that } \quad 35 \times 37 \\ =(30+5)(30+7) \\=30^{2}+(5+7) 30+5 \times 7 \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b ] \\ =900+360+35=1295\\ \end{array}\begin{array}{l} \text {(iv) Given that } \quad 53 \times 55 \\ =(50+3)(50+5) \\=50^{2}+(3+5) 50+3 \times 5 \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b ] \\ =2500+400+15=2915\\ \end{array}\begin{array}{l} \text {(v) Given that } \quad 103 \times 96 \\ =(100+3)(100-4) \\ =100^{2}+(3-4) 100-3 \times 4 \quad \quad [\text {Using: } (x+a)(x-b)=x^{2}+(a-b) x-a b] \\ =10000-100-12=9888\\ \end{array}\begin{array}{l} \text {(vi) Given that } \quad 34 \times 36 \\ =(30+4)(30+6) \\=30^{2}+(4+6) 30+4 \times 6 \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b ] \\=900+300+24=1224\\ \end{array}\begin{array}{l} \text {(vii) Given that } \quad 994 \times 1006 \\=(1000-6) x(1000+6) \\=1000^{2}+(6-6) \times 1000-6 \times 6 \quad \quad [\text {Using: } (x+a)(x+b)=x^{2}+(a+b) x+a b] \\ =1000000-36=999964 \\ \end{array}