# Problems on Factorization of Expressions of the Form a$$^{2}$$ - b$$^{2}$$

Here we will solve different types of Problems on Factorization of expressions of the form a2 – b2.

1. Resolve into factors: 49a2 – 81b2

Solution:

Given expression = 49a2 – 81b2

= (7a)2 – (9b)2

= (7a + 9b)(7a – 9b).

2. Factorize: (x + y)2 – 4(x – y)2

Solution:

Given expression = (x + y)2 – 4(x – y)2

= (x + y)2 – {2(x – y)}2

= {(x + y) + 2(x – y)}{(x + y) - 2(x – y)}

= (x + y + 2x – 2y)(x + y - 2x + 2y)

= (3x – y)(3y - x).

3. Factorize the expression (x2 + y2 – z2)2 – 4x2y2 of the form a2 – b2.

Solution:

Given expression = (x$$^{2}$$ + y$$^{2}$$ – z$$^{2}$$)$$^{2}$$ – 4x$$^{2}$$y$$^{2}$$

= (x$$^{2}$$ + y$$^{2}$$ – z$$^{2}$$)$$^{2}$$ – (2xy)$$^{2}$$

= (x$$^{2}$$ + y$$^{2}$$ – z$$^{2}$$ + 2xy)(x$$^{2}$$ + y$$^{2}$$ – z$$^{2}$$ – 2xy)

= (x$$^{2}$$ + 2xy + y$$^{2}$$ - z$$^{2}$$)(x$$^{2}$$ – 2xy + y$$^{2}$$ – z$$^{2}$$)

= {(x$$^{2}$$ + 2xy + y$$^{2}$$) - z$$^{2}$$}{(x$$^{2}$$ - 2xy + y$$^{2}$$) – z$$^{2}$$}

= {(x + y)$$^{2}$$ - z$$^{2}$$}{(x -  y)$$^{2}$$ - z$$^{2}$$}

= (x + y + z)(x + y - z)(x -  y + z) (x -  y - z).

4. Factorize 2x$$^{2}$$ - 18 of the form a$$^{2}$$ – b$$^{2}$$.

Solution:

Given expression = 2x$$^{2}$$ - 18

= 2(x$$^{2}$$ – 9)

= 2(x$$^{2}$$ – 3$$^{2}$$)

= 2(x + 3)(x - 3)

5. Factorize: 25(a + b)$$^{2}$$ – (a – b)$$^{2}$$

Solution:

Given expression = 25(a + b)$$^{2}$$ – (a – b)$$^{2}$$

= {5(a + b)}$$^{2}$$ – (a – b)$$^{2}$$

= {5(a + b) + (a – b)}{5(a + b) – (a – b)}

= (5a + 5b + a – b)(5a + 5b - a + b)

= (6a + 4b)(4a + 6b)

= {2(3a + 2b)}{2(2a + 3b)}

= 4(3a + 2b)(2a + 3b)

6. Factorize the expression 9(x + y)$$^{2}$$ – x$$^{2}$$ of the form a$$^{2}$$ – b$$^{2}$$.

Solution:

Given expression = 9(x + y)$$^{2}$$ – x$$^{2}$$

= {3(x + y)}$$^{2}$$ – x$$^{2}$$

= {3(x + y) + x}{3(x + y) - x}

= (3x + 3y + x)(3x + 3y - x)

= (4x + 3y)(2x + 3y)

7. Factorize the expression 9x$$^{2}$$ – 4(y + 2x)$$^{2}$$ of the form a$$^{2}$$ – b$$^{2}$$.

Solution:

Given expression = 9x$$^{2}$$ – 4(y + 2x)$$^{2}$$

= (3x)$$^{2}$$ – {2(y + 2x)}$$^{2}$$

= {3x + 2(y + 2x)}{3x - 2(y + 2x)}

= (3x + 2y + 4x)(3x - 2y - 4x)

= (7x + 2y)(-x - 2y)

= (7x + 2y){-(x + 2y)}

= -(7x + 2y)(x + 2y)

8. Factorize: 1 – (b – c)$$^{2}$$

Solution:

Given expression = 1 – (b – c)$$^{2}$$

= 1$$^{2}$$ – (b – c)$$^{2}$$

= {1 + (b – c)}{1 - (b – c)}

= (1 + b – c)(1 - b + c)

9. Factorize: 81a$$^{4}$$ – 16b$$^{4}$$

Solution:

Given expression = 81a$$^{4}$$ – 16b$$^{4}$$

= (9a$$^{2}$$)$$^{2}$$ – (4b$$^{2}$$)$$^{2}$$

= (9a$$^{2}$$ + 4b$$^{2}$$)(9a$$^{2}$$ - 4b$$^{2}$$)

= (9a$$^{2}$$ + 4b$$^{2}$$){(3a)$$^{2}$$ - (2b)$$^{2}$$}

= (9a$$^{2}$$ + 4b$$^{2}$$)(3a + 2b)(3a - 2b)

10. Factorize: 16ax$$^{4}$$ – ay$$^{4}$$

Solution:

Given expression = 16ax$$^{4}$$ – ay$$^{4}$$

= a(16x$$^{4}$$ – y$$^{4}$$)

= a{(4x$$^{2}$$)$$^{2}$$ – (y$$^{2}$$)$$^{2}$$}

= a(4x$$^{2}$$ + y$$^{2}$$)(4x$$^{2}$$ - y$$^{2}$$)

= a(4x$$^{2}$$ + y$$^{2}$$){(2x)$$^{2}$$ - y$$^{2}$$}

= a(4x$$^{2}$$ + y$$^{2}$$)(2x + y)(2x – y)

11. Factorize: a$$^{4}$$ – 16b$$^{4}$$

Solution:

Given expression = a$$^{4}$$ – 16b$$^{4}$$

= (a$$^{2}$$)$$^{2}$$ – (4b$$^{2}$$)$$^{2}$$

= (a$$^{2}$$ + 4b$$^{2}$$)(a$$^{2}$$ - 4b$$^{2}$$)

= (a^2 + 4b$$^{2}$$){a$$^{2}$$ – (2b)$$^{2}$$}

= (a^2 + 4b$$^{2}$$)(a + 2b)(a - 2b)

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