Quadratic equations (Short cut Method)

Type:-

** (x+ 7)(x+ 9)= (x +3)(x+ 21).

Here 7x9= 3x21

Therefore x= 0

  EXERCISE -1

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1) (x+ 2)(x+ 10)= (x +5)(x+ 4).

2) (x+ 1)(x+ 10)= (x +2)(x+ 5).

3) (x+ 12)(x+ 9)= (x + 54)(x+ 2).

4) (x+ 12)(x+ 4)= (x +6)(x+ 8).

5) (x+ 20)(x+ 5)= (x +50)(x+ 2).

6) (x+ 21)(x+ 2)= (x +7)(x+ 6).

7) (x+ 18)(x+ 2)= (x +4)(x+ 9).

8) (x+ 700)(x+ 90)= (x +3)(x+ 2100).

9) (x+ 15)(x+ 6)= (x +3)(x+ 30).

10) (x+ 1)(x+ 34)= (x +2)(x+ 17).

TYPE -2;

** 1/(2x -1) + 1/(3x -1)= 0.

Here (2x-1)+(3x- 1)=0

  5x - 2= 0

  x= 2/5

EXERCISE- 2

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1) 1/(3x -2) + 1/(2x -1)= 0.

2) 1/(7x -2) + 1/(3x - 5)= 0.

3) 1/(8x - 5) + 1/(13x - 7)= 0.

4) 1/(2x + 3) + 1/(3x + 7)= 0.

5) 1/(x +11) + 1/(x + 12)= 0.

6) 1/(x - 21) + 1/(x - 31)= 0.

7) 1/(2x +9) + 1/(3x +23)= 0.

8) 1/(12x +1) + 1/(3x - 8)= 0.

9) 1/(222x - 7) + 1/(31x - 9)= 0.

10) 1/(87x - 61) + 1/(3x - 9)= 0.

11) 1/(23x + 69) + 1/(42x -1)= 0.

12) 1/(202x - 89) + 1/(32x -19)= 0.

13) 1/(202x - 89) + 1/(32x -19)= 0.

14) 3/(2x - 89) + 3/(3x -9)= 0.

15) 5/(5x - 2) + 5/(x +19)= 0.

16) 1/(3 - 4x) + 1/(7 - 2x) = 0.

17) 1/(1 - 9x) + 1/(8 - 3x) = 0.

18) 1/(2 - 5x) + 1/(7 + 6x) = 0.

19) 1/(30 - 9x) + 1/(2 - 8x) = 0.

20) 1/(7 + 3x) + 1/(9 - x) = 0.

21) 1/(54 - 4x) + 1/(89 - x) = 0.

22) 1/(2021 - x) + 1/(2022 - x) = 0.

23) 1/(309 - 5x) + 1/(709 - 8x) = 0.

24) 6/(1 - 3x) + 6/(2 - 2x) = 0.

25) 1/(67 - 4x) + 1/(67 - 2x) = 0.

EXERCISE-3

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*** (2x+9)/(2x +7)= (2x+7)/(2x +9).

N' + N" = D' + D"

Then (2x +9)+(2x+7)= 0

Or, 4x +16= 0

Or, x= -4

1) (3x+4)/(6x +5)= (4x+9)/(x +8).

2) (x+ 9)/(6x +5)= (20x+9)/(15x +13).

3) (13x+ 1)/(6x +5)= (14x+9)/(26x +5).

4) (7x+41)/(6x +5)= (41x+9)/(42x +45).

5) (x+41)/(3x +1)= (4x+9)/(2x +49).

6) (2x+ 7)/(6x +5)= (8x+9)/(4x +11).

7) (301x+4)/(205x +5)= (4x+9)/(100x +8).

8) (3x+4)/(6x +7)= (x+ 1)/(2x + 3).

9) (3x+4)/(6x +7)= (5x+6)/(2x +3).

EXERCISE -4

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** 1/(x - 7) + 1/(x -9)= 1/(x - 6) + 1/(x - 10).

D' + D" = D"' + D""

2x -16= 0

x= 8

1) 1/(x +7) + 1/(x +9)= 1/(x + 6) + 1/(x +10).

2) 1/(x - 7) + 1/(x +9)= 1/(x +11) + 1/(x - 9).

3) 1/(x - 8) + 1/(x -9)= 1/(x - 5) + 1/(x - 12).

4) 1/(x - 8) - 1/(x - 5)= 1/(x - 12) - 1/(x - 9).

5) 1/(x + 1) - 1/(x +3)= 1/(x +2) - 1/(x + 4).

6) 1/(x + 1) - 1/(x - 3)= 1/(x - 4) - 1/(x - 8).

7) 1/(x - b) - 1/(x - b - d)= 1/(x - c + d) - 1/(x - c).

8) 1/(x + b) - 1/(x + b + d)= 1/(x + c - d) - 1/(x + c).

9) (x - 2)/(x - 3) + (x - 3)/(x - 4)= (x - 1)/(x - 2) + (x - 4)/(x - 5).

10) x /(x - 2) + (x - 9)/(x - 7)= (x + 1)/(x - 1) + (x - 8)/(x - 6).

11) (2x - 3)/(x - 2) + (3x - 20)/(x - 7)= (x - 3)/(x - 4) + (4x - 19)/(x - 5).

12) (3x - 8)/(x - 3) + (4x - 35)/(x - 9)= (2x - 9)/(x - 5) + (5x - 34)/(x - 7).

13) (3x - 13)/(x - 4) + (4x - 41)/(x - 10)= (2x - 13)/(x - 6) + (5x - 42)/(x - 8).

14) (4x + 21)/(x +5) + (5x - 69)/(x - 14)= (3x - 5)/(x - 2) + (6x - 41)/(x - 7).

15) (x²+ 3x +3)/(x +2) + (x² - 15)/(x - 4)= (x²+ 7x +11)/(x +5) + (x² - 4x - 20)/(x - 7).

16) 2/(2x + 3) + 3/(3x +2) = 1/(x + 1) + 6/(6x +7).

17) 3/(3x + 1) - 6/(6x +1) = 3/(3x + 2) - 2/(2x +1).

18) 3/(3x + 1) + 2/(2x -1) = 3/(3x - 2) + 2/(2x +1).

19) 1/(x + 3) + 3/(3x -1) = 1/(x + 5) + 3/(3x -7).

20) (2x +11)/(x + 5) - (9x - 9)/(3x - 4) = (4x +13)/(x + 3) - (15x - 47)/(3x - 10).

21) (5 - 6x)/(3x - 1) + (2x +7)/(x +3) = (31- 12x)/(3x -7) + (4x +21)/(x +5).

EXERCISE - 5

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1) x + 1/x = 26/5.                    5, 1/5

2) x/(x+1) + (x +1)/x = 82/9.      1/8, -9/8

3) (x +1)/(x+2) + (x +2)/(x +1) = 37/6.          -11/5, -4/6

4) (x +4)/(x - 4) + (x - 4)/(x +4) = 10/3         8, - 8

5) x + 1/x = 13/6.               3/2, 2/3

6) x+ 1/x = 25/12.            4/3, 3/4

7) (x +5)/(x+ 6) + (x + 6)/(x +5) = 29/10.          -13/3, - 20/3

8) x /(x+ 1) + (x + 1)/x = 169/60.          5/7, - 12/7

9) (2x +11)/(2x -11) + (2x -11)/(2x +11) = 193/84.            

10) x - 1/x = 45/14.             7/2, -2/7

11) x /(x+ 3) -  (x + 3)/x = 15/56.          - 24, - 7/5

12) (x + 7)/(x+ 9) - (x + 9)/(x + 7) = 32/63.          -16, - 63/8

13) (5x +9)/(5x - 9) - (5x -9)/(5x +9) = 56/45          63/10, - 18/35

EXERCISE - 6

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*** (2x +6)/(3x+5) = (5x +2)/(4x +3). 

Sum of N = Sum of D

Then, 7x +10= 0 or -3x +4= 0

1) (3x +4)/(6x+7) = (5x +6)/(2x +3).   -5/4, -1

2) (7x +5)/(9x -5) = (9x +7)/(7x +17).   -3/4, 5

3) (3x -9)/(2x- 9) = (9x -7)/(14x -7).   1, 0

4) (16x -3)/(7x+7) = (2x -15)/(11x -25).   1, 10/9

EXERCISE-7

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