COMPLEX ROOTS OF QUADRATIC EQUATION WORKSHEET

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

Answers

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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