Verified answer

Questions:-

Factorize the following (any 10):-

(1) x² + 8x - 20

(2) x² - 6x - 16

(3) x² - 6x + 8

(4) x² - 5x - 50

(5) x² - 2x - 63

(6) x² - 7x + 10

(7) x² + 17x + 72

(8) x² + 4x - 60

(9) x² - 18x + 80

(10) x² + 5x + 6

Solutions:-

(1) x² + 8x - 20

⟶ By splitting middle term,

= x² + 10x - 2x - 20

Taking common terms out,

= x(x + 10) - 2(x + 10)

Again taking common,

= (x + 10)(x - 2)

Ans: (x + 10)(x - 2)

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(2) x² - 6x - 16

⟶ x² - 8x + 2x - 16

= x(x - 8) + 2(x - 8)

= (x - 8)(x + 2)

Ans: (x - 8)(x + 2)

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(3) x² - 6x + 8

⟶ x² - 4x - 2x + 8

= x(x - 4) - 2(x - 4)

= (x - 4)(x - 2)

Ans: (x - 4)(x - 2)

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(4) x² - 5x - 50

⟶ x² - 10x + 5x - 50

= x(x - 10) + 5(x - 10)

= (x - 10)(x + 5)

Ans: (x - 10)(x + 5)

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(5) x² - 2x - 63

⟶ x² - 2x - 63

= x² - 9x + 7x - 63

= x(x - 9) + 7(x - 9)

= (x - 9)(x + 7)

Ans: (x - 9)(x + 7)

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(6) x² - 7x + 10

⟶ x² - 5x - 2x + 10

= x(x - 5) - 2(x - 5)

= (x - 5)(x - 2)

Ans: (x - 5)(x - 2)

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(7) x² + 17x + 72

⟶ x² + 8x + 9x + 72

= x(x + 8) + 9(x + 8)

= (x + 8)(x + 9)

Ans: (x + 8)(x + 9)

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(8) x² + 4x - 60

⟶ x² + 10x - 6x - 60

= x(x + 10) - 6(x + 10)

= (x + 10)(x - 6)

Ans: (x + 10)(x - 6)

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(9) x² - 18x + 80

⟶ x² - 10x - 8x + 80

= x(x - 10) - 8(x - 10)

= (x - 10)(x - 8)

Ans: (x - 10)(x - 8)

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(10) x² + 5x + 6

⟶ x² + 2x + 3x + 6

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

= (x + 2)(x + 3)

Ans: (x + 2)(x + 3)

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Solution :

1. x² + 8x - 20

> x² + 10x - 2x - 20

> x( x + 10) - 2( x + 10)

> ( x - 2)( x + 10)

2. x² - 6x - 16

> x² - 8x + 2x - 16

> x( x - 8) + 2( x - 8)

> ( x + 2)( x - 8)

3. x² - 6x + 8

> x² - 4x - 2x + 8

> x( x - 4) + 2( x - 4)

> ( x + 2)( x - 4)

4. x² - 5x - 50

> x² - 10x + 5x - 50

> x( x - 10) + 5( x - 10)

> ( x + 5)( x - 10)

5 . x² - 2x - 63

> x² - 9x + 7x - 63

> x( x - 9) + 7( x - 9)

> ( x + 7)( x - 9)

6. x² - 7x + 10

> x² - 5x - 2x + 10

> x( x - 5) - 2( x - 5)

> ( x - 2)( x - 5)

7. x² + 17x + 72

> x² + 14x + 3x + 72

> x( x + 14) + 3( x + 14)

> ( x + 3)( x + 14)

8. x² + 4x - 60

> x² + 10x - 6x - 60

> x( x + 10) - 6( x + 10)

> ( x - 6)( x + 20)

9. x² + 18x + 80

> x² + 10x + 8x + 80

> x( x + 10) + 8( x + 10)

> ( x + 8)( x + 10)

10. x² + 5x + 6

> x² + 2x + 3x + 6

>x( x + 2) + 3( x + 2)

> ( x + 2)( x + 3)

11. 10x³ - 19x² - 61x - 8

> 10x³ - 35x² + 16x² - 5x - 56x - 8

> 5x ( 2x² - 7x - 1) + 8( 2x² - 7x + 1)

> ( 5x + 8)( 2x² - 7x - 1)

12. 20x³ - 15x² + 28x - 21 [ Correction ; If its +15 x², can't be solved ]

> 20x³ + 28x - 15x² - 21

> 4x ( 5x² + 7) - 3( 5x² + 7)

> ( 4x - 3)( 5x² + 7)

13 . 36x³ - 5x² - 46x - 5

> 36x³ - 9x² - 45x + 4x² - x - 5

> 9x ( 4x² - x - 5) + ( 4x² - x - 5)

> ( 9x + 1)( 4x² - x - 5)

> ( 9x + 1)( 4x² + 4x - 5x - 5)

> ( 9x + 1)[ 4x( x + 1) - 5( x + 1) ]

> ( 9x + 1)( 4x - 5)( x + 1)

14. 10ab + 4a + 5b + 2

> 2a( 5b + 2) + ( 5b + 2)

> ( 2a + 2)( 5b + 2)

15. 2x² - 32

> 2( x² - 16)

> 2( x + 5)( x - 4)

16. x² - 7x + 12

> x² - 3x - 4x + 12

>x( x - 3) - 4( x - 3)

> (x - 3)( x - 4)

17. 2x² + 7x + 6

> 2x² + 3x + 4x + 6

> x( 2x + 3) + 2( 2x + 3)

> ( x + 2)( 2x + 3)

18. x² - 6x - 40

> x² - 10x + 4x - 40

> x( x - 10) + 4( x - 10)

> ( x + 4)( x - 10)

19. 2x² - 15x + 22

> 2x² - 4x - 11x + 22

> 2x( x - 2) - 11( x - 2)

> (x - 2)( 2x - 11)

20. 8x² - 21x + 10

> 8 x² - 16x - 5x + 10

> 8x( x - 2) - 5( x - 2)

> ( x - 2)( 8x - 5)

21. 25 x² - 20 xy + 4y²

> [ 5x ]² - 2[ 5x ][ 2y] + [ 2y ]²

> [ 5x - 2y ]²

22. 4x² + 4xy + y²

> [ 2x ]² + 2[ 2x ][ y ] + y².

> [ 2x + y ]².

23. 6x² - 11xy - 10y²

> 6x² - 15xy + 4xy - 10y²

> 3x ( 2x - 5y) + 2y( 2x - 5y)

> ( 3x + 2y)( 2x - 5y)

24. 4x² + 17x + 15

>4x² + 12x + 5x + 15

> 4x ( x + 3) + 5( x + 3)

> (x + 3)( 4x + 5)

25. 8x² - 22x + 15

> 8x² - 10x - 12x + 15

> 2x( 4x - 5) - 3( 4x - 5)

> (2x - 3)( 4x - 5)

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Additional information -

(a + b)² = a² + 2ab + b²

(a + b)² = (a - b)² + 4ab

(a - b)² = a² - 2ab + b²

(a - b)² = (a + b)² - 4ab

a² + b² = (a + b)² - 2ab

a² + b² = (a - b)² + 2ab

2 (a² + b²) = (a + b)² + (a - b)²

4ab = (a + b)² - (a - b)²

ab = {(a + b)/2}² - {(a-b)/2}²

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

(a + b)³ = a³ + 3a²b + 3ab² b³

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - 3a²b + 3ab² - b³

a³ + b³ = (a + b)( a² - ab + b² )

a³ + b³ = (a + b)³ - 3ab( a + b)

a³ - b³ = (a - b)( a² + ab + b²)

a³ - b³ = (a - b)³ + 3ab ( a - b )

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