SOLVING QUADRATICS REAL AND IMAGINARY SOLUTIONS

To solve quadratic equation, we use the following methods.

(1)  Factoring

(2)  Quadratic formula

(3)  Completing the square method

Without solving the quadratic equation completely, we can decide whether it has real and not equal solution, real and equal solution or imaginary solutions.

Using the 

Find roots of the quadratic equations given below.

Example 1 :

x²-6x+12  =  0

Solution :

x²-6x+12  =  0

a  =  1, b  =  -6 and c  =  12

Solving the quadratic equation using the formula

x  =  -b±√(b²-4ac)/2a

we can solve the equation.

b²-4ac  =  6² - 4 ⋅ 1 ⋅ 12

=  36 - 48

=  -12 < 0

Since b²-4ac < 0, the quadratic equation will have imaginary roots.

√b²-4ac  =  √-12  ==>  2i√3

x  =  (6 ± 2i√3)/2

x  =  (3 ± i√3)

So, the roots are 3 + i√3 and 3 - i√3.

Example 1 :

14-3x²  =  2x

Solution :

14-3x²  =  2x

Multiply by negative.

3x² - 14  =  -2x

3x²+2x-14  =  0

a  =  3, b  =  2 and c  =  -14

Solving the quadratic equation using the formula

x  =  -b±√(b²-4ac)/2a

we can solve the equation.

b²-4ac  =  2² - 4 ⋅ 3 ⋅ (-14)

=  4 + 168

=  172 > 0

Since b²-4ac > 0, the quadratic equation will have real roots.

√b²-4ac  =  √172  ==>  2√43

x  =  (2 ± 2√43)/2

x  =  (1 ± √43)

So, the roots are (1+√43)/2 and (1-√43)/2.+

Find a possible pair of integer values for a and c so that the quadratic equation has the given solution(s). 

Example 3 :

ax²+4x+c  =  0 has two imaginary solution.

Solution :

Since the quadratic equation has imaginary roots.

b²-4ac < 0

a  =  a, b  =  4 and c  =  c

4²-4ac < 0

16-4ac < 0

To  get the imaginary  solutions, 4ac should be > 16

So, product of a and c should be > 4

One of the possible value If a  =  2 and b  =  3

So, the required equation is 2x²+4x+3  =  0

Example 4 :

 ax² − 8x + c = 0 two real solutions

Solution :

Since the quadratic equation has imaginary roots.

b²-4ac > 0

a  =  a, b  =  -8 and c  =  c

(-8)²-4ac > 0

64-4ac > 0

To  get the real solutions, 4ac should be > 64

So, product of a and c should be < 16

One of the possible value If a  =  2 and b  =  3

So, the required equation is 2x² − 8x + 3 = 0

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