6.5. Solved Problems

1. Solve: 2x + 5 = 9

   2x + 5 - 5 = 9 - 5
   
   2x = 4
   
    x = 2
   

2. Solve: 4x + 3 = -17

   4x + 3 - 3 = -17 - 3
   
   4x = -20
   
    x = -5 
   

3. Solve: 2.5x - 3.2 = 9.3

   2.5x - 3.2 + 3.2 = 9.3 + 3.2
   
   2.5x = 12.5 
   
      x = 5
   

4. Solve: y² + 2y - 8 = 0

   (y + 4)(y - 2) = 0
   
   y + 4 = 0 or y - 2 = 0
   
   y = -4, 2
   

5. Solve: (t + 3)^1/2 = t - 3

   t + 3 = (t - 3)2 = t2 - 6t + 9
   
   0 = t2 - 7t + 6
   
   0 = (t - 1)(t - 6)
   
   t = 1, 6 are possible solutions. Check,
   
   t = 1, (4)1/2 ≠ -2 --> is extraneous
   
   t = 6, (9)1/2 =  3 --> is a solution
   

6. Solve: x² + 48 = 0

         _____      _
   x = ±√ -48  = ±4√3i
   

7. Solve: x² + 10x + 5 = 0

   x2 + 10x + 25 = 20    complete square of lhs
   
   (x + 5)2 = 20         factor
              _
   x + 5 = ±2√5
              _
   x = -5 ± 2√5
   

8. Solve: x² + 14x - 2 = 0

   x2 + 14x + 49 = 51    complete square of lhs
   
   (x + 7)2 = 51         factor
             __
   x + 7 = ±√51
             __
   x = -7 ± √51
   

9. Solve: 3x² + 6x + 7 = 0

              __________          _______
        -6 ± √62 - 4×3×7    -6 ± √36 - 84 
   x = —————————————————— = —————————————— =
             2×3                   6
              _____            _             _
        -6 ± √ -48      -6 ± 4√3i     -3 ± 2√3i 
   x = ————————————— = ——————————— = ———————————
             6              6             3
   

10. Solve: x² + 3x + 6 = 0

               ________           _____
         -3 ± √32 - 4×6     -3 ± √ -15
    x = ———————————————— = ————————————— = 
               2                 2
               __
         -3 ± √15i
    x = ————————————
              2
    

11. Solve: x² - 4x + 13 = 0

              _______          _____
         4 ± √16 - 52     4 ± √ -36      4 ± 6i
    x = —————————————— = ———————————— = ———————— =
               2               2           2
    
    x = 2 ± 3i
    

12. Solve: 2x + 3 + 2(3x - 2) = 2(4x - 1) + 1

    2x + 3 + 6x - 4 = 8x - 2 + 1
    
    8x - 1 = 8x - 1
    
    This equation is an identity;
    it is true for all values of the variable x.
    

13. Solve: 3(3x - 2) = 2(5x - 1) - x + 2

    9x - 6 = 10x - 2 - x + 2
    
    9x - 6 = 9x 
    
    -6 = 0 
    This equation has no solution; 
    it is not true for any value of the variable x.
    

14. Solve: 6/(x + 2) = 2 - 3x/(x + 2)

    6 = 2(x + 2) - 3x
    
    6 = 2x + 4 - 3x = -x + 4
    
    x = -2
    
    This equation has no solution; 
    if x = -2, the equation has undefined terms.
    

15. Solve: 1 = 1/(x + 1) + 2/(x + 2)

    x2 + 3x + 2 = (x + 2) + 2(x + 1)
    
    x2 + 3x + 2 = 3x + 4
    
    x2 = 2
          _
    x = ±√2
    

16. Solve: x⁴ - 7x² - 8 = 0

    Let x2 = u, then substitute:
    
    u2 - 7u - 8 = 0
    
    (u + 1)(u - 8) = 0
    
    u = -1 or u = 8
    
    Then,
    
    x2 = -1 or x2 = 8 
    
    So the four solutions are:
                     _
    x = ±i or x = ±2√2
    

17. Solve: x^2/3 - 5x^1/3 + 6 = 0

    Let x1/3 = u, then substitute:
    
    u2 - 5u + 6 = 0
    
    (u - 2)(u - 3) = 0
    
    u = 2 or u = 3
    
    Then,
    
    x1/3 = 2 or x1/3 = 3
    
    x = 8 or x = 27
    

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

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

19. Solve for x:

     1     1     1     1
    ——— = ——— + ——— + ———
     x     a     b     c
    
    abc = bcx + acx + abx = x(ac + ab + bc)
    
             abc
    x = ——————————————
         ac + ab + bc
    

20. Solve: x³ - 8 = 0

    x3 - 23 = (x - 2)(x2 + 2x + 4)
    
    x = 2 or
               ________           _____            _
         -2 ± √ 4 - 16      -2 ± √ -12      -2 ± 2√3i 
    x = ———————————————— = ————————————— = ————————————— 
              2                 2               2     
              _
      = -1 ± √3i
                                             _
    There are three solutions:  x = 2, -1 ± √3i