Check whether the following quadratic equations have complex roots. If so, solve the given quadratic equation and find the two complex roots.
Problem 1 :
x2 - x + 1 = 0
Problem 2 :
x2 + 3x + 5 = 0
Problem 3 :
x2 - 5x + 6 = 0
Problem 4 :
x2 + 6x + 9 = 0
Problem 5 :
x - 2 = -5/x
Problem 6 :
3x2 + 10x + 9 = 0
Problem 7 :
-x + 3 = 2/(x - 2)
Problem 8 :
x2/2 = 3x - 5
1. Answer :
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.
2. Answer :
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.
3. Answer :
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.
4. Answer :
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.
5. Answer :
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.
6. Answer :
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.
7. Answer :
-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.
8. Answer :
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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