Class 8 Exam > Class 8 Notes > Mathematics (Maths) Class 8 > Factorization - Exercise 7.9
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Page 1
Q u e s t i o n : 1 7 9
Factorize each of the following quadratic polynomials by using the method of completing the square:
p
2
+ 6p + 8
S o l u t i o n :
p2+6p+8=p2+6p+622-622+8 [Adding and subtracting 622, that is, 32]=p2+6p+32-32+8=p2+2×p×3+32-9+8=p2+2×p×3+32-1=(p+3)2-12 [Completing the square]=[(p+3)-1]
=(p+3-1)(p+3+1)=(p+2)(p+4)
Q u e s t i o n : 1 8 0
Factorize each of the following quadratic polynomials by using the method of completing the square:
q2 - 10q + 21
S o l u t i o n :
q2-10q+21=q2-10q+1022-1022+21 [Adding and subtracting 1022, that is, 52]=q2-2×q×5+52-52+21=(q-5)2-4 [Completing the square]=(q-5)2-22 =[(q-5)-2][(q-5)+2]=(q
=(q-7)(q-3)
Q u e s t i o n : 1 8 1
Factorize each of the following quadratic polynomials by using the method of completing the square:
4y
2
+ 12y + 5
S o l u t i o n :
4y2+12y+5=4(y2+3y+54) [Making the coefficient of y2=1]=4[y2+3y+322-322+54] [Adding and subtracting 322]=4[(y+32)2-94+54]=4[(y+32)2-12]
=4[(y+32)-1][(y+32)+1]=4(y+32-1)(y+32+1)=4(y+12)(y+52)=(2y+1)(2y+5)
Q u e s t i o n : 1 8 2
Factorize each of the following quadratic polynomials by using the method of completing the square:
p
2
+ 6p - 16
S o l u t i o n :
p2+6p-16=p2+6p+622-622-16 [Adding and subtracting 622, that is, 32]=p2+6p+32-9-16=(p+3)2-25 [Completing the square]=(p+3)2-52=[(p+3)-5][(p+3)+5]=(p+3-5)(p+3+5
=(p-2)(p+8)
Q u e s t i o n : 1 8 3
Factorize each of the following quadratic polynomials by using the method of completing the square:
x
2
+ 12x + 20
S o l u t i o n :
x2+12x+20=x2+12x+1222-1222+20 [Adding and subtracting 1222, that is, 62]=x2+12x+62-62+20=(x+6)2-16 [Completing the square]=(x+6)2-42=[(x+6)-4][(x+6
=(x+6-4)(x+6+4)=(x+2)(x+10)
Q u e s t i o n : 1 8 4
Factorize each of the following quadratic polynomials by using the method of completing the square:
a
2
- 14a - 51
S o l u t i o n :
a2-14a-51=a2-14a+1422-1422-51 [Adding and subtracting 1422, that is, 72]=a2-14a+72-72-51=(a-7)2-100 [Completing the square]=(a-7)2-102 =[(a-7)-10][(a-7)+10]=(a
=(a-17)(a+3)
Q u e s t i o n : 1 8 5
Factorize each of the following quadratic polynomials by using the method of completing the square:
a
2
+ 2a - 3
S o l u t i o n :
a2+2a-3=a2+2a+222-222-3 [Adding and subtracting 222, that is, 12]=a2+2a+12-12-3=(a+1)2-4 [Completing the square]=(a+1)2-22=[(a+1)-2][(a+1)+2]=(a+1-2)(a+1+2)=
Q u e s t i o n : 1 8 6
Factorize each of the following quadratic polynomials by using the method of completing the square:
4x
2
- 12x + 5
S o l u t i o n :
4x2-12x+5=4(x2-3x+54) [Making the coefficient of x2=1]=4[x2-3x+322-322+54] [Adding and subtracting 322]=4[(x-32)2-94+54] [Completing th
=4[(x-32)2-12] =4[(x-32)-1][(x-32)+1]=4(x-32-1)(x-32+1)=4(x-52)(x-12)=(2x-5)(2x-1)
Q u e s t i o n : 1 8 7
Factorize each of the following quadratic polynomials by using the method of completing the square:
y
2
- 7y + 12
S o l u t i o n :
y2-7y+12=y2-7y+722-722+12 [Adding and subtracting 722]=(y-72)2-494+484 [Completing the square]=(y-72)2-14 =(y-72)2-122 =[(y-72)-12][(y-72)+12]=(y-72-12)(y-72+12)=(y-4
Page 2
Q u e s t i o n : 1 7 9
Factorize each of the following quadratic polynomials by using the method of completing the square:
p
2
+ 6p + 8
S o l u t i o n :
p2+6p+8=p2+6p+622-622+8 [Adding and subtracting 622, that is, 32]=p2+6p+32-32+8=p2+2×p×3+32-9+8=p2+2×p×3+32-1=(p+3)2-12 [Completing the square]=[(p+3)-1]
=(p+3-1)(p+3+1)=(p+2)(p+4)
Q u e s t i o n : 1 8 0
Factorize each of the following quadratic polynomials by using the method of completing the square:
q2 - 10q + 21
S o l u t i o n :
q2-10q+21=q2-10q+1022-1022+21 [Adding and subtracting 1022, that is, 52]=q2-2×q×5+52-52+21=(q-5)2-4 [Completing the square]=(q-5)2-22 =[(q-5)-2][(q-5)+2]=(q
=(q-7)(q-3)
Q u e s t i o n : 1 8 1
Factorize each of the following quadratic polynomials by using the method of completing the square:
4y
2
+ 12y + 5
S o l u t i o n :
4y2+12y+5=4(y2+3y+54) [Making the coefficient of y2=1]=4[y2+3y+322-322+54] [Adding and subtracting 322]=4[(y+32)2-94+54]=4[(y+32)2-12]
=4[(y+32)-1][(y+32)+1]=4(y+32-1)(y+32+1)=4(y+12)(y+52)=(2y+1)(2y+5)
Q u e s t i o n : 1 8 2
Factorize each of the following quadratic polynomials by using the method of completing the square:
p
2
+ 6p - 16
S o l u t i o n :
p2+6p-16=p2+6p+622-622-16 [Adding and subtracting 622, that is, 32]=p2+6p+32-9-16=(p+3)2-25 [Completing the square]=(p+3)2-52=[(p+3)-5][(p+3)+5]=(p+3-5)(p+3+5
=(p-2)(p+8)
Q u e s t i o n : 1 8 3
Factorize each of the following quadratic polynomials by using the method of completing the square:
x
2
+ 12x + 20
S o l u t i o n :
x2+12x+20=x2+12x+1222-1222+20 [Adding and subtracting 1222, that is, 62]=x2+12x+62-62+20=(x+6)2-16 [Completing the square]=(x+6)2-42=[(x+6)-4][(x+6
=(x+6-4)(x+6+4)=(x+2)(x+10)
Q u e s t i o n : 1 8 4
Factorize each of the following quadratic polynomials by using the method of completing the square:
a
2
- 14a - 51
S o l u t i o n :
a2-14a-51=a2-14a+1422-1422-51 [Adding and subtracting 1422, that is, 72]=a2-14a+72-72-51=(a-7)2-100 [Completing the square]=(a-7)2-102 =[(a-7)-10][(a-7)+10]=(a
=(a-17)(a+3)
Q u e s t i o n : 1 8 5
Factorize each of the following quadratic polynomials by using the method of completing the square:
a
2
+ 2a - 3
S o l u t i o n :
a2+2a-3=a2+2a+222-222-3 [Adding and subtracting 222, that is, 12]=a2+2a+12-12-3=(a+1)2-4 [Completing the square]=(a+1)2-22=[(a+1)-2][(a+1)+2]=(a+1-2)(a+1+2)=
Q u e s t i o n : 1 8 6
Factorize each of the following quadratic polynomials by using the method of completing the square:
4x
2
- 12x + 5
S o l u t i o n :
4x2-12x+5=4(x2-3x+54) [Making the coefficient of x2=1]=4[x2-3x+322-322+54] [Adding and subtracting 322]=4[(x-32)2-94+54] [Completing th
=4[(x-32)2-12] =4[(x-32)-1][(x-32)+1]=4(x-32-1)(x-32+1)=4(x-52)(x-12)=(2x-5)(2x-1)
Q u e s t i o n : 1 8 7
Factorize each of the following quadratic polynomials by using the method of completing the square:
y
2
- 7y + 12
S o l u t i o n :
y2-7y+12=y2-7y+722-722+12 [Adding and subtracting 722]=(y-72)2-494+484 [Completing the square]=(y-72)2-14 =(y-72)2-122 =[(y-72)-12][(y-72)+12]=(y-72-12)(y-72+12)=(y-4
Q u e s t i o n : 1 8 8
Factorize each of the following quadratic polynomials by using the method of completing the square:
z
2
- 4z - 12
S o l u t i o n :
z2-4z-12=z2-4z+422-422-12 [Adding and subtracting 422, that is, 22]=z2-4z+22-22-12=(z-2)2-16 [Completing the square]=(z-2)2-42=[(z-2)-4][(z-2)+4]=(z-6)(z+2)
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