Class 8 RD Sharma Solutions Chapter 7 Factorization Exercise 7.9 (original) (raw)
Last Updated : 12 Sep, 2024
Exercise 7.9 in Chapter 7 of RD Sharma's Class 8 Mathematics textbook represents the culmination of the factorization techniques covered throughout the chapter. This exercise is designed to challenge students with a comprehensive mix of factorization problems, requiring them to apply and combine various methods they've learned. The questions in this set often involve more complex algebraic expressions, sometimes requiring multiple steps or the application of several factorization techniques in sequence.
Factorize each of the following quadratic polynomials by using the method of completing the square
**Question 1. p 2 + 6p + 8
**Solution:
p2 + 6p + 8
= p2 + 2 x p x 3 + 32 – 32 + 8 (completing the square)
= (p2 + 6p + 32) – 1
= (p + 3)2 – 1
= (p + 3)2 – (1)2 { ∵ a2 + b2 = (a + b) (a - b)}
= (p + 3 + 1) (p + 3 - 1)
= (p + 4) (p + 2)
**Question 2. q 2 – 10q + 21
**Solution:
q2 – 10q + 21
= (q)2 – 2 x q x 5 + (5)2 – (5)2 + 21 (completing the square)
= (q)2 – 2 x q x 5 + (5)2 - 25+ 21
= (q)2 - 2 x q x 5 + (5)2 – 25 +21
= (q)2 - 2 x q x 5 + (5)2 – 4
= (q – 5)2 – (2) {∵ a2 – b2 = (a + b) (a – b)}
= (q - 5 + 2) (q - 5 - 2)
=(q - 3) (q - 7)
**Question 3. 4y 2 + 12y + 5
**Solution:
4y2 + 12y + 5
= (2y)2 + 2 x 2y x 3 + (3)2 – (3)2 + 5 (completing the square)
= (2y + 3)2 – 9 + 5
= (2y + 3)2 – 4
= (2y + 3)2 - (2)2 {∵ a2 – b2 = (a + b) (a – b)}
= (2y + 3 + 2) (2y + 3 – 2)
= (2y + 5) (2y + 1)
**Question 4. p 2 + 6p - 16
**Solution:
p2 + 6p – 16
= (p)2 + 2 x p x 3 + (3)2 – (3)2 – 16 (completing the square)
= (p)2 + 2 x p x 3 + (3)2 – 9 – 16
= (p + 3)2 – 25
= (p + 3)2 – (5)2 {∵ a2 - b2 = {a + b) (a – b)}
= (p + 3 + 5)(p + 3 - 5)
= (p + 8) (p – 2)
**Question 5. x 2 + 12x + 20
**Solution:
x2 + 12x + 20
= (x)2 + 2 x **x x 6 + (6)2 – (6)2 + 20 (completing the square)
= (x)2 + 2 x **x x 6 + (6)2 -36 + 20
= (x + 6)2 -16
= (x + 6)2 – (4)2 {∵ a2 – b2 = (a + b) (a – b)}
= (x + 6 + 4) (x + 6 – 4)
= (x + 10) (x + 2)
**Question 6. a 2 – 14a – 51
**Solution:
a2 – 14a - 51
= (a)2 – 2 x a x 7 + (7)2 – (7)2 – 51 (completing the square)
= (a)2 – 2 x a x 7 + (7)2 – 49 – 51
= (a – 7)2 – 100
= (a – 7)2 – (10)2 {∵ a2 – b2 = (a + b) (a – b)}
= (a – 7 + 10) (a – 7 – 10)
= (a + 3) (a – 17)
**Question 7. a 2 + 2a – 3
**Solution:
a2 + 2a – 3
= (a)2 + 2 x a x 1 + (1)2 – (1)2 – 3 (completing the square)
= (a)2 + 2 x a x 1 + (1)2 – 1 – 3
= (a + 1)2 – 4
= (a + 1)2 – (2)2 {∵ a2 – b2 = (a + b) (a – b)}
= (a + 1 + 2) (a + 1 – 2)
= (a + 3) (a – 1)
**Question 8. 4x 2 – 12x + 5
**Solution:
4x2 – 12x + 5
= (2x)2 – 2 x **2x x 3 + (3)2 – (3)2 + 5 (completing the square)
= (2x)2 – 2 x **2x x 3 + (3)2 - 9 + 5
= (2x – 3)2 – 4
= (2x – 3)2 – (2)2 {∵ a2 – b2 = (a + b) (a – b)}
= (2x – 3 + 2) (2x – 3 – 2)
= (2x – 1) (2x – 5)
**Question 9. y 2 – 7y + 12
**Solution:
y2 – 7y + 12
= (y)2 – 2 × y × 7/2 + (7/2)2 - (7/2)2 + 12 (completing the square)
= (y)2 – 2 × y × 7/2 + 49/4 - 49/4 + 12
= (y - 7/2)2 - (49 - 48)/4
= (y - 7/2)2 - 1/4
= (y - 7/2)2 - (1/2)2 {∵ a2 – b2 = (a + b) (a - b)}
= (y - 7/2 + 1/2) (y - 7/2 - 1/2)
= (y - 6/2) (y - 8/2)
= (y - 3) (y - 4)
**Question 10. z 2 **- 4z -12
**Solution:
z2 – 4z – 12
= (z)2 – 2 x z x 2 + (2)2 – (2)2 – 12 (completing the square)
= (z)2 – 2 x z x 2 + (2)2 – 4 – 12
= (z - 2)2 - 16
= (z - 2)2- (4)2 {∵ a2 – b2 = (a + b) (a – b)}
= (z – 2 + 4) (z – 2 – 4)
= (z + 2)(z - 6)
Summary
Exercise 7.9 serves as a capstone to the factorization chapter, presenting students with a diverse array of problems that test their mastery of all previously covered techniques. The questions range from factoring polynomials with multiple variables to handling expressions that combine various algebraic structures like perfect squares, cubes, and difference of squares. This exercise emphasizes the importance of recognizing patterns, selecting appropriate factorization methods, and applying them systematically. By successfully navigating these challenges, students not only reinforce their factorization skills but also develop critical thinking and problem-solving abilities that are crucial for more advanced mathematical concepts they will encounter in future studies.