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Solution of Quadratic Equations when b2 - 4ac < 0

In this case, i.e., when b2-4ac<0, the roots or solutions are complex.
 

Example 1

Consider 2x2-2x+5=0      a=2 b= -2 c=5

b2-4ac =(-2)2 -4 * 2* 5

           =4-40

         =-36<0

The roots will be complex.

Using the formula
 

Example 2

Solve the quadratic equation

-7x2+13x-15

a=-7,   b=13,   c=-15

Discriminant Δ=b2-4ac
                       = (13)2- 4 *-7 *-15
                       = 169 - 420
                       = -351 < 0

Roots are complex.
Use the formula to solve the equation
 
 

Try these questions

To solve the quadratic equations

1.

x2-x+5 = 0

2.

x2+4x+5 = 0

3.

2x2-2x+5 = 0

4.

9x2+12x = -29

5.

49x2-70x = -34

6.

x2+22x+122 = 0

7.

x2+x = -1

8.

3x2+7x+20 = 0

 

Answers to Practice Problems

1.

To solve x2-x+5=0
a=1 b=-1 c=5
Discriminant Δ= b2-4ac
                      = (-1)2-4*1*5
                      = 1- 20
                      = -19< 0
Roots are complex.

Using the formula to solve the equation

   
 
 

2.

To solve x2+4x+5=0
a=1, b=4, c=5
Discriminant Δ=b2-4ac
                      =(4)2-4*1*5
                      =16-20
                      =-4<0
Roots are complex.

Using the formula to solve the equation

   
 
  x = -2 ± i
Roots are x=-2+i, -2-i
 

3.

To solve 2x2-2x++5 = 0
a=2 b=-2 c=5
Discriminant Δ= b2-4ac
                       = (-2)2-4*2*5
                       = 4-40
                       = -36 < 0
The roots are complex.
Using the formula to solve the equations

   
 
 

4.

To solve   9x2+12x = -29
Transposing 9x2+12x+2a = 0
a=9, b=12, c=29

Discriminant Δ= b2-4ac
                      = (12)2-4*9*29
                      = 144-1044
                      = -900< 0
Roots are complex. Using the formula to solve the equations

   
 
 

5.

To solve 49x2-70x = -34
Transposing 49x2-70x+34 = 0
a=49, b=-70, c=34
Discriminant Δ= b2-4ac
                       = (-70)2-4*49*34
                       = 4900-6664
                       = -1764< 0
Roots are complex. Using the formula to solve the equations

   
 
 

6.

To solve x2-22x+122 = 0
a=1, b=22, c=122
Discriminant Δ= b2-4ac
                       = (22)2-4*1*122
                       = 484-488
                       = -4< 0
Roots are complex. Using the formula to solve the equations

   
 
 

7.

To solve x2+x = -1

Trasposing x2+x+1 = 0
a=1, b=1, c=1

Discriminant Δ= b2-4ac
                       = (1)2-4*1*1
                       = 1-4
                       = -3< 0
Roots are complex. Using the formula to solve the equation

   
 
 

8.

To solve 3x2+7x+20 = 0
a=3, b=7, c= 20
Discriminant Δ= b2- 4ac
                       = (7)2- 4 * 3 * 20
                       = 49-240
                       = -191< 0
Roots are complex. Using the formula to solve the equations

   
 

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