## Worksheet on complex roots

In this worksheet on complex roots we are going to see problems in 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.

Quadratic equation:  ax² +bx+c=0, a≠0. 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:

Re writing the quadratic equation

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:

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

Worksheet on complex roots

Find the roots of the given quadratic equations:

1.  x² - x + 1 = 0

(A) x = (1 ± i√3)/2

(B) x = 1, 2

(C) x = 1 ± i

2.  x² + 3x + 5 = 0

(A) x+ √2, x-√2

(B) (-3±i√11)/2

(C) -x±i√7

3.  -5/x = x-2

(A) x+ √2, x-√2

(B) (-3±i√11)/2

(C) x±4i

4.  3x²+ 10x + 9 =0

(A) (-5 ± i√2)/3

(B) (-3±i√11)/2

(C) x±4i

5.  (-x+3) = 2/(x-2)

(A) (-5 ± i√2)/3

(B) (-3±i√11)/2

(C) (5±i√7)/2

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