## Complex roots

Complex roots:  While solving quadratic equations, we get sometimes unreal roots(imaginary roots).

While solving quadratic equations, we will find the value of the discriminant to find the nature of the roots.

 Discriminant ∆= b² - 4 ac (1) ∆ > 0 but not a perfect square (2) ∆ > 0 but a perfect square (3) ∆ = 0 (4) ∆ < 0 but a perfect square Nature of roots Real,unequal and irrational Real,unequal and rational Real, equal and rational Complex roots

Examples:

1.  Solve the equation :

x²/2 = 3x - 5

Solution:

x² = 2(3x-5)

x² =  6x -10

x² - 6x +10 = 0.

a = 1,  b = -6, c = 10

The value of x is

x  =  -(-6) ± √[(-6)² - 4(1)(10)]

2(1)

x  =   [6 ± √(36-40)]/2

=  [6 ± √(-4)]/2

=   (6 ± 2i)/2

=    3 ± i

So       x   =   3 + i, or 3-i

2.  Find the roots of the equation:

x² - 2 x + 5 = 0

Solution: In this equation

a = 1, b = -2, c =5

So substituting in the formula

x  =  -(-2) ± √[(-2)² - 4 (1)(5)]/ 2(1)

=    ( 2  ± √ (4-20)) / 2

=     ( 2 ± √(-16)) / 2

=      (2 ±  4i) / 2

=        1 ± 2i

So   x  =    1+ 2i or 1-2i

3.  Find the roots of the equation:

x + 5/x = 3

Solution:

Rewriting the equation,

x(x + 5/x )  = 3x

x²  +  5        =  3x

x² - 3x + 5   =  0

Here a = 1,  b = -3, c = 5.

So substituting in the formula

x  =  -(-3) ± √[(-3)² - 4 (1)(5)]/ 2(1)

=    ( 23 ± √ (9-20)) / 2

=     ( 3 ± √(-11)) / 2

=      (3 ± √11 i) / 2

So        x  =    ( 3 + √11 i)/2  or  ( 3 - √11 i)/2

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