COMPACT (Quadratic Equation, Cubic Equations, Biquadratic Equations) Short Cut Method

TYPE -1

1) x + 1/x = 17/4.                       4 or 1/4

2) x+ 1/x = 26/5.                       5 or 1/5

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

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

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

Type-2

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

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

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

4) (2x+11)/(2x-11) + (2x-11)/(2x+11)= 193/84.                  

Type -3

1) x - 1/x =5/6.                             3/2,-2/3

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

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

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

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


Type -4

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

2) (7x +5)/(8x-5) = (9x+7)/(7x+17) .    -3/4 , 5

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

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


Type -5

1) 2/(x+2) + 3/(x+3)= 4/(x+4) + 1/(x+1).    -5/2, 9   
                                    
2) 3/(x+3) + 4/(x+4)= 2/(x+2) + 5/(x+5).      0,  -7/2            

3) 1/(2x+1) + 1/(3x+1)= 2/(3x+2) + 1/(6x+1).    0, -5/12               

4) a/(x+a) + b/(x+b)= (a- c)/(x+a- c) + (b+ c)/(x+b+ c).    0, -1/2(a+ 2b + c)


Type-6

Continue.......                  




Cubic Equations

1) x³- 6x²+11x - 6=0.                       1,2,3

2) x³+ 6x²+11x + 6=0.                 -3,-2,-1

3) x³+ 6x²-37x +30=0.                 1,3,-10

4) x³+ 9x²+23x +15=0.               -5,-3,-1

5) x³+ 9x²+24x +16=0.                      -1,-4

6) x³+ 7x²+14x +8=0.              -2,+4,±1

7) x³+ 8x²+17x +10=0.           -2,-1-5

8) x³+ 10x²+27x +18=0.               -3,-6,-1

**9) x³ - 7x +6=0.   


Biquadratic Equations

1) x⁴+ 4x³ - 25x²- 16x +84=0.    -2,2,3,-7

2) x⁴+ 8x³ +14x²- 8x -15=0.      -1,-3,1,-5

3) x⁴-12x³ +49x²- 78x +40 =0.   1,2,4,5

4) x⁴+ 16x³ +86x²+176x +105=0.      -1, -3, -5, -7

5) x⁴- 16x³ +91x²- 216x +180=0.    2,3,5,6

6) x⁴- 20x³ +137x²- 382x +360=0.      2,4, 5, 9
                                      

                           

                             


 


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