COMPLEX ROOTS OF A QUADRATIC EQUATION
Complex Number
A complex number is a combination of a real number and an imaginary number.
In Math, when there is negative sign inside the square root, it is considered to be an imaginary number.
Examples :
√-3, √-4
We know that 2 is a real number √-3 is an imaginary number. Then, 3 + √-4 and 3 - √-4 are complex numbers.
In imaginary numbers like √-3 and √-4, we take i2 for -1.
Then,
|
√-3 = √[3(-1)] = √(3i2) = i√3 |
√-4 = √[4(-1)] = √(22i2) = 2i |
So, the complex numbers 3 + √-4 and 3 - √-4 can be written in terms of i.
|
3 + √-4 = 3 + 2i |
3 - √-4 = 3 - 2i |
Solving a Quadratic Equations Using Quadratic Formula
We can use quadratic formula to solve a quadratic equation which is in standard form, that is
ax2 + bx + c = 0
Quadratic Formula :
In the quadratic formula above, if the value of b2 - 4ac is negative or
b2 - 4ac < 0,
then the quadratic equation will have complex roots.
Check whether the following quadratic equations have complex roots. If so, solve the given quadratic equation and find the two complex roots.
Example 1 :
x2 - x + 1 = 0
Solution :
x2 - x + 1 = 0
Comparing ax2 + bx + c = 0 and x2 - x + 1 = 0, we get
a = 1, b = -1 and c = 1
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = (-1)2 - 4(1)(1)
= 1 - 4
= -3 < 0
Since b2 - 4ac < 0, the given quadratic equation has complex roots.
Using Quadratic formula to solve the given quadratic equation and find the two complex roots.
Quadratic Formula :
Substitute a = 1, b = -1 and c = 1.
Example 2 :
x2 + 3x + 5 = 0
Solution :
x2 + 3x + 5 = 0
Comparing ax2 + bx + c = 0 and x2 + 3x + 5 = 0, we get
a = 1, b = 3 and c = 5
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = 32 - 4(1)(5)
= 9 - 20
= -11 < 0
Since b2 - 4ac < 0, the given quadratic equation has complex roots.
Using Quadratic formula to solve the given quadratic equation and find the two complex roots.
Quadratic Formula :
Substitute a = 1, b = 3 and c = 5.
Example 3 :
x2 - 5x + 6 = 0
Solution :
x2 - 5x + 6 = 0
Comparing ax2 + bx + c = 0 and x2 - 5x + 6 = 0, we get
a = 1, b = -5 and c = 6
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = (-5)2 - 4(1)(6)
= 25 - 24
= 1 > 0
Since b2 - 4ac > 0, the given quadratic equation does not have complex roots.
Example 4 :
x2 + 6x + 9 = 0
Solution :
x2 + 6x + 9 = 0
Comparing ax2 + bx + c = 0 and x2 + 6x + 9 = 0, we get
a = 1, b = 6 and c = 9
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = 62 - 4(1)(9)
= 36 - 36
= 0
Since b2 - 4ac = 0, the given quadratic equation does not have complex roots.
Note : Only if b2 - 4ac < 0, the quadratic equation will have complex roots.
Example 5 :
x - 2 = -5/x
Solution :
x - 2 = -5/x
Multiply both sides by x.
x2 - 2x = -5
Add 5 to both sides.
x2 - 2x + 5 = 0
Comparing ax2 + bx + c = 0 and x2 - 2x + 5 = 0, we get
a = 1, b = -2 and c = 5
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = (-2)2 - 4(1)(5)
= 4 - 20
= -16 < 0
Since b2 - 4ac < 0, the given quadratic equation has complex roots.
Using Quadratic formula to solve the given quadratic equation and find the two complex roots.
Quadratic Formula :
Substitute a = 1, b = -2 and c = 5.
Example 6 :
3x2 + 10x + 9 = 0
Solution :
3x2 + 10x + 9 = 0
Comparing ax2 + bx + c = 0 and 3x2 + 10x + 9 = 0, we get
a = 3, b = 10 and c = 9
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = 102 - 4(3)(9)
= 100 - 108
= -8 < 0
Since b2 - 4ac < 0, the given quadratic equation has complex roots.
Using Quadratic formula to solve the given quadratic equation and find the two complex roots.
Quadratic Formula :
Substitute a = 3, b = 10 and c = 9.
Example 7 :
-x + 3 = 2/(x - 2)
Solution :
-x + 3 = 2/(x - 2)
Multiply both sides by (x - 2).
(-x + 3)(x - 2) = 2
-x2 + 2x + 3x - 6 = 2
-x2 + 5x - 6 = 2
Subtract 2 from both sides.
-x2 + 5x - 8 = 0
Multiply both sides by -1.
x2 - 5x + 8 = 0
Comparing ax2 + bx + c = 0 and x2 - 5x + 8 = 0, we get
a = 1, b = -5 and c = 8
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = (-5)2 - 4(1)(8)
= 25 - 32
= -7 < 0
Since b2 - 4ac < 0, the given quadratic equation has complex roots.
Using Quadratic formula to solve the given quadratic equation and find the two complex roots.
Quadratic Formula :
Substitute a = 1, b = -5 and c = 8.
Example 8 :
x2/2 = 3x - 5
Solution :
x2/2 = 3x - 5
Multiply both sides by 2.
x2 = 2(3x - 5)
x2 = 6x - 10
x2 - 6x + 10 = 0
Comparing ax2 + bx + c = 0 and x2 - 6x + 10 = 0, we get
a = 1, b = -6 and c = 10
Find the value of the discriminant b2 - 4ac.
b2 - 4ac = (-6)2 - 4(1)(10)
= 36 - 40
= -4 < 0
Since b2 - 4ac < 0, the given quadratic equation has complex roots.
Using Quadratic formula to solve the given quadratic equation and find the two complex roots.
Quadratic Formula :
Substitute a = 1, b = -6 and c = 10.
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