Class 8 Factorization Exercise 7.6-3

\begin{array}{l} \text {Q21. Factorize the algebraic expression} \quad x^{2}-y^{2}+6 y-9\\ \\\text {Sol. We have } x^{2}-y^{2}+6 y-9 \\ =x^{2}- (y^{2} -6 y + 9) \\ =x^{2}- (y^{2} -2 \times 3 \times y + 3^{2}) \\ =x^{2} -(y-3)^{2} \quad [\text {Using : } (a -b)^{2} = a^{2}+b^{2}-2 a b] \\ =(x+y-3)(x-y+3) \quad [\text {Using : } a^{2} – b^{2} = (a+b)(a-b)] \\ \\\text {Q22. Factorize the algebraic expression} \quad 25 x^{2}-10 x+1-36 y^{2}\\ \\\text {Sol. We have } 25 x^{2}-10 x+1-36 y^{2} \\ =[(5 x)^{2}-2(5 x)+1]-(6 y)^{2} \\ =(5 x-1)^{2}-(6 y)^{2} \quad [\text {Using : } (a -b)^{2} = a^{2}+b^{2}-2 a b] \\ =(5 x-6 y-1)(5 x+6 y-1) \quad [\text {Using : } a^{2} – b^{2} = (a+b)(a-b)] \\ \\\text {Q23. Factorize the algebraic expression} \quad a^{2}-b^{2}+2 b c-c^{2}\\ \\\text {Sol. We have } a^{2}-b^{2}+2 b c-c^{2} \\ =a^{2}-(b^{2}-2 b c+c^{2}) \\ =a^{2}-(b-c)^{2} \quad [\text {Using : } (a -b)^{2} = a^{2}+b^{2}-2 a b] \\ =(a+b-c)(a-b+c) \quad [\text {Using : } a^{2} – b^{2} = (a+b)(a-b)] \\ \\\text {Q24. Factorize the algebraic expression} \quad a^{2}+2 a b+b^{2}-c^{2}\\ \\\text {Sol. We have } a^{2}+2 a b+b^{2}-c^{2} \\ =(a+b)^{2}-c^{2} \quad [\text {Using : } (a + b)^{2} = a^{2}+b^{2} + 2 a b] \\ (a+b+c)(a+b-c) \quad [\text {Using : } a^{2} – b^{2} = (a+b)(a-b)] \\ \\\text {Q25. Factorize the algebraic expression} \quad 49-x^{2}-y^{2}+2 x y\\ \\\text {Sol. We have } 49-x^{2}-y^{2}+2 x y \\ =7^{2}-(x^{2}+y^{2}-2 x y) \\ =7^{2}-(x-y)^{2} \quad [\text {Using : } (a – b)^{2} = a^{2}+b^{2} – 2 a b] \\ [7+(x-y)][7-(x-y)] \quad [\text {Using : } a^{2} – b^{2} = (a+b)(a-b)] \\ = (x-y+7)(y-x+7)\\ \\\text {Q26. Factorize the algebraic expression} \quad a^{2}+4 b^{2}-4 a b-4 c^{2}\\ \\\text {Sol. We have } a^{2}+4 b^{2}-4 a b-4 c^{2} \\ =[a^{2}-2(a)(2 b)+(2 b)^{2}]-(2 c)^{2} \\ =(a-2 b)^{2}-(2 c)^{2} \quad [\text {Using : } (a – b)^{2} = a^{2}+b^{2} – 2 a b] \\ =(a-2 b+2 c)(a-2 b-2 c) \quad [\text {Using : } a^{2} – b^{2} = (a+b)(a-b)] \\ \\\text {Q27. Factorize the algebraic expression} \quad x^{2}-y^{2}-4 x z+4 z^{2}\\ \\\text {Sol. We have } x^{2}-y^{2}-4 x z+4 z^{2} \\ =[x^{2}-2(x)(2 z)+(2 z)^{2}]-y^{2} \\ =(x-2 z)^{2}-y^{2} \quad [\text {Using : } (a – b)^{2} = a^{2}+b^{2} – 2 a b] \\ =(x-2 z +y)(x-2 z -y) \quad [\text {Using : } a^{2} – b^{2} = (a+b)(a-b)] \\ =(x+y-2 z)(x-y-2 z) \end{array}