Worksheet on Factoring the Differences of Two Squares

Worksheet on factoring the differences of two squares will help us to factorize an algebraic expression using the following identity a² - b² = (a + b) (a – b).

1. Factorize the following by taking the difference of squares:

(i) x² – 9

(ii) a² – 1

(iii) 49 – x²

(iv) 4x² – 25

(v) a²b² – 16

(vi) a⁴ – b⁴



2. Factoring by the Difference of Two Perfect Squares:

(i) 144a² – 169b²

(ii) 1 – 0.09a²

(iii) 16x² – 121

(iv) – 64a² + (9/25) b²

(v) x⁴ – 256

(vi) (x + y)⁴ – z⁴



3. Factorize using the formula of differences of two squares:

(i) 36a² – b2

(ii) x²y² – 16

(iii) 9a⁴b⁴ – 25p⁴q⁴

(iv) x⁴ – 256

(v) 81x² – 49y²

(vi) x² – (y – z)²



4. Factor the difference of two perfect squares:

(i) 16m² – (3n + 2y)²

(ii) (3a + 4b)² - (4b + 5b)²

(iii) (x + y)² - (x – y)²

(iv) 50p² – 72q²

(v) a⁴ - (b + c)⁴

(vi) m² - 1/169



5. Factor each expression as a difference between two squares: (i) 9 (x + y)² - 4 (x – y)²

(ii) 16/49 - 25p²

(iii) 9xy² – x³

(iv) 4 (3x + 1)² - 9 (x – 2)²

(v) 1 - 121a²

(vi) 169p² - 1

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6. Factor using the identity:

(i) 1 - (a + b)²

(ii) x²y² - 25/z²

(iii) x¹²y⁴ - x⁴y¹²

(iv) 100 (x – y)² – 121 (a + b)²

(v) 2x – 50x³

(vi) 25/x² - (4x²)/9

(vii) x⁴ - 1/(y⁴)

(viii) 75x³y² – 108xy⁴

Answers for the worksheet on factoring the differences of two squares are given below to check the exact answers of the above factorization.

Answers:

1. (i) (x + 3) (x - 3)  

(ii) (a + 1) (a - 1)   

(iii) (7 + x) (7 - x)   

(iv) (2x + 5) (2x - 5)

(v) (ab + 4) (ab - 4)  

(vi) (a² + b²) (a + b) (a - b)

2. (i) (12a + 13b) (12a - 13b) 

(ii) (1 + 0.3a) (1 - 0.3a)  

(iii) (4x + 11) (4x - 11)   

(iv) [(3/5)b + 8a] [(3/5)b - 8a]  

(v) (x² + 16) (x + 4) (x – 4)

(vi) [(x + y)² + z²] (x + y + z) (x + y - z)

3. (i) (6a + b) (6a - b)                                                   

(ii) (xy + 4) (xy - 4)

(iii) (3a²b² + 5p²q²) (3a²b² - 5p²q²)

(iv) (x² + 16) (x + 4) (x - 4)

(v) (9x + 7y) (9x - 7y) 

(vi) (x + y - z) (x – y + z)     

4. (i) (4m + 3n + 2y) (4m - 3n - 2y)   

(ii) (3a  + 8b + 5d) (3a – 5d)  

(iii) 4xy   

(iv) 2(5p + 6q) (5p - 6q)

(v) (a² + b² + c² + 2bc) (a + b + c) (a – b - c)

(vi) (m + 1/13) (m - 1/13)

5. (i) (5x + y) (x + 5y)    

(ii) (4/7 + 5p) (4/7 - 5p)     

(iii) x(3y + x) (3y - x)      

(iv) (9x – 4) (3x + 8)  

(v) (1 + 11a) (1 - 11a)    

(vi) (13p + 1) (13p - 1)  

6. (i) (1 + a + b) (1 – a - b)

(ii) (xy + 5/z) (xy - 5/z)

(iii) x⁴y⁴ (x⁴ + y⁴) (x² + y²) (x + y) (x – y)

(iv) (10x - 10y + 11a + 11b) (10x - 10y - 11a - 11b)

(v) 2x (1 + 5x) (1 - 5x)

(vii) (x² + 1/y² ) (x + 1/y) (x - 1/y)

(viii) 3xy² (5x + 6y) (5x - 6y)

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