RD Sharma solution class 8 chapter 7 Factorization Exercise 7.9

Exercise 7.9

Page-7.32

Question 1:

Factorize each of the following quadratic polynomials by using the method of  completing the square:
p² + 6p + 8

Answer 1:

$$\begin{gathered} {\mathrm{p}}^2+6\mathrm{p}+8 \\ ={\mathrm{p}}^2+6\mathrm{p}+{\left(\frac{6}{2}\right)}^2-{\left(\frac{6}{2}\right)}^2+8 [\mathrm{Adding} \mathrm{and} \mathrm{subtracting} {\left(\frac{6}{2}\right)}^2, \mathrm{that} \mathrm{is}, 3^2] \\ ={\mathrm{p}}^2+6\mathrm{p}+3^2-3^2+8 \\ ={\mathrm{p}}^2+2\times \mathrm{p}\times 3+3^2-9+8 \\ ={\mathrm{p}}^2+2\times \mathrm{p}\times 3+3^2-1 \\ =(\mathrm{p}+3{)}^2-1^2 [\mathrm{Completing} \mathrm{the} \mathrm{square}] \\ =[(\mathrm{p}+3)-1\left]\right[(\mathrm{p}+3)+1] \\ =(\mathrm{p}+3-1\left)\right(\mathrm{p}+3+1) \\ =(\mathrm{p}+2\left)\right(\mathrm{p}+4) \end{gathered}$$

Question 2:

Factorize each of the following quadratic polynomials by using the method of completing the square:
q2 − 10q + 21

Answer 2:

$$\begin{gathered} {\mathrm{q}}^2-10\mathrm{q}+21 \\ ={\mathrm{q}}^2-10\mathrm{q}+{\left(\frac{10}{2}\right)}^2-{\left(\frac{10}{2}\right)}^2+21 [\mathrm{Adding} \mathrm{and} \mathrm{subtracting} {\left(\frac{10}{2}\right)}^2, \mathrm{that} \mathrm{is}, 5^2] \\ ={\mathrm{q}}^2-2\times \mathrm{q}\times 5+5^2-5^2+21 \\ =(\mathrm{q}-5{)}^2-4 [\mathrm{Completing} \mathrm{the} \mathrm{square}] \\ =(\mathrm{q}-5{)}^2-2^2 \\ =\left[\right(\mathrm{q}-5)-2]\left[\right(\mathrm{q}-5)+2] \\ =(\mathrm{q}-5-2)(\mathrm{q}-5+2) \\ =(\mathrm{q}-7)(\mathrm{q}-3) \end{gathered}$$

Question 3:

Factorize each of the following quadratic polynomials by using the method of completing the square:
4y² + 12y + 5

Answer 3:

$$\begin{gathered} 4y^2+12y+5 \\ =4(y^2+3y+\frac{5}{4}) [Making the coefficient of y^2=1] \\ =4[y^2+3y+{\left(\frac{3}{2}\right)}^2-{\left(\frac{3}{2}\right)}^2+\frac{5}{4}] [Adding and subtracting {\left(\frac{3}{2}\right)}^2] \\ =4\left[\right(y+\frac{3}{2}{)}^2-\frac{9}{4}+\frac{5}{4}] \\ =4[(y+\frac{3}{2}{)}^2-1^2] [Completing the square] \\ =4\left[\right(y+\frac{3}{2})-1]\left[\right(y+\frac{3}{2})+1] \\ =4(y+\frac{3}{2}-1)(y+\frac{3}{2}+1) \\ =4(y+\frac{1}{2})(y+\frac{5}{2}) \\ =(2y+1)(2y+5) \end{gathered}$$

Question 4:

Factorize each of the following quadratic polynomials by using the method of completing the square:
p² + 6p − 16

Answer 4:

$$\begin{gathered} {\mathrm{p}}^2+6\mathrm{p}-16 \\ ={\mathrm{p}}^2+6\mathrm{p}+{\left(\frac{6}{2}\right)}^2-{\left(\frac{6}{2}\right)}^2-16 [\mathrm{Adding} \mathrm{and} \mathrm{subtracting} {\left(\frac{6}{2}\right)}^2, \mathrm{that} \mathrm{is}, 3^2] \\ ={\mathrm{p}}^2+6\mathrm{p}+3^2-9-16 \\ =(\mathrm{p}+3{)}^2-25 [\mathrm{Completing} \mathrm{the} \mathrm{square}] \\ =(\mathrm{p}+3{)}^2-5^2 \\ =\left[\right(\mathrm{p}+3)-5]\left[\right(\mathrm{p}+3)+5] \\ =(\mathrm{p}+3-5)(\mathrm{p}+3+5) \\ =(\mathrm{p}-2)(\mathrm{p}+8) \end{gathered}$$

Question 5:

Factorize each of the following quadratic polynomials by using the method of completing the square:
x² + 12x + 20

Answer 5:

$$\begin{gathered} x^2+12x+20 \\ =x^2+12x+{\left(\frac{12}{2}\right)}^2-{\left(\frac{12}{2}\right)}^2+20 [Adding and subtracting {\left(\frac{12}{2}\right)}^2, that is, 6^2] \\ =x^2+12x+6^2-6^2+20 \\ =(x+6{)}^2-16 [Completing the square] \\ =(x+6{)}^2-4^2 \\ =\left[\right(x+6)-4]\left[\right(x+6)+4] \\ =(x+6-4)(x+6+4) \\ =(x+2)(x+10) \end{gathered}$$

Question 6:

Factorize each of the following quadratic polynomials by using the method of completing the square:
a² − 14a − 51

Answer 6:

$$\begin{gathered} a^2-14a-51 \\ =a^2-14a+{\left(\frac{14}{2}\right)}^2-{\left(\frac{14}{2}\right)}^2-51 [Adding and subtracting {\left(\frac{14}{2}\right)}^2, that is, 7^2] \\ =a^2-14a+7^2-7^2-51 \\ =(a-7{)}^2-100 [Completing the square] \\ =(a-7{)}^2-{10}^2 \\ =\left[\right(a-7)-10]\left[\right(a-7)+10] \\ =(a-7-10)(a-7+10) \\ =(a-17)(a+3) \end{gathered}$$

Page-7.33

Question 7:

Factorize each of the following quadratic polynomials by using the method of completing the square:
a² + 2a − 3

Answer 7:

$$\begin{gathered} a^2+2a-3 \\ =a^2+2a+{\left(\frac{2}{2}\right)}^2-{\left(\frac{2}{2}\right)}^2-3 [Adding and subtracting {\left(\frac{2}{2}\right)}^2, that is, 1^2] \\ =a^2+2a+1^2-1^2-3 \\ =(a+1{)}^2-4 [Completing the square] \\ =(a+1{)}^2-2^2 \\ =\left[\right(a+1)-2]\left[\right(a+1)+2] \\ =(a+1-2)(a+1+2) \\ =(a-1)(a+3) \end{gathered}$$

Question 8:

Factorize each of the following quadratic polynomials by using the method of completing the square:
4x² − 12x + 5

Answer 8:

$$\begin{gathered} 4x^2-12x+5 \\ =4(x^2-3x+\frac{5}{4}) [Making the coefficient of x^2=1] \\ =4[x^2-3x+{\left(\frac{3}{2}\right)}^2-{\left(\frac{3}{2}\right)}^2+\frac{5}{4}] [Adding and subtracting {\left(\frac{3}{2}\right)}^2] \\ =4\left[\right(x-\frac{3}{2}{)}^2-\frac{9}{4}+\frac{5}{4}] [Completing the square] \\ =4[(x-\frac{3}{2}{)}^2-1^2] \\ =4\left[\right(x-\frac{3}{2})-1]\left[\right(x-\frac{3}{2})+1] \\ =4(x-\frac{3}{2}-1)(x-\frac{3}{2}+1) \\ =4(x-\frac{5}{2})(x-\frac{1}{2}) \\ =(2x-5)(2x-1) \end{gathered}$$

Question 9:

Factorize each of the following quadratic polynomials by using the method of completing the square:
y² − 7y + 12

Answer 9:

$$\begin{gathered} y^2-7y+12 \\ =y^2-7y+{\left(\frac{7}{2}\right)}^2-{\left(\frac{7}{2}\right)}^2+12 [Adding and subtracting {\left(\frac{7}{2}\right)}^2] \\ =(y-\frac{7}{2}{)}^2-\frac{49}{4}+\frac{48}{4} [Completing the square] \\ =(y-\frac{7}{2}{)}^2-\frac{1}{4} \\ =(y-\frac{7}{2}{)}^2-{\left(\frac{1}{2}\right)}^2 \\ =[(y-\frac{7}{2})-\frac{1}{2}\left]\right[(y-\frac{7}{2})+\frac{1}{2}] \\ =(y-\frac{7}{2}-\frac{1}{2}\left)\right(y-\frac{7}{2}+\frac{1}{2}) \\ =(y-4\left)\right(y-3) \end{gathered}$$

Question 10:

Factorize each of the following quadratic polynomials by using the method of completing the square:
z² − 4z − 12

Answer 10:

$$\begin{gathered} z^2-4z-12 \\ =z^2-4z+{\left(\frac{4}{2}\right)}^2-{\left(\frac{4}{2}\right)}^2-12 [Adding and subtracting {\left(\frac{4}{2}\right)}^2, that is, 2^2] \\ =z^2-4z+2^2-2^2-12 \\ =(z-2{)}^2-16 [Completing the square] \\ =(z-2{)}^2-4^2 \\ =\left[\right(z-2)-4]\left[\right(z-2)+4] \\ =(z-6)(z+2) \end{gathered}$$