Worksheet on Nature of the Roots of a Quadratic Equation | Nature of Roots of Quadratic Equation Worksheet with Answers

Are you lagging in solving the nature of roots of the quadratic equation? If your answer is yes, then don’t get tensed we are here to help you to find the Nature of roots of a quadratic equation in simple techniques. Acquire the math skills to solve the algebraic problems with the help of a Worksheet on Nature of the Roots of a Quadratic Equation with answers. We can solve the nature of the roots by determining the discriminant D (Δ) = b² – 4ac

  • If D (Δ) = b² – 4ac = 0 the roots are real, rational and equal.
  • If D (Δ) = b² – 4ac > 0 the roots are real, rational, and unequal
  • If D (Δ) = b² – 4ac < 0 the roots are non-real, imaginary

See More:

Worksheet on Nature of the Roots of a Quadratic Equation

Let us discuss in detail the nature of the roots of a quadratic equation with some examples.

Example 1.
Find the discriminant and nature of roots of the quadratic equation x² – 6x + 9 = 0.

Solution:

D = b² – 4ac
a = 1, b = -6, c = 9
= (-6)² – 4(1)(9)
= 36 – 36
= 0
D = 0
Thus the roots are real, rational and equal.


Example 2.
What is the sum and product of the Roots if the Quadratic Equation is x² – 5x + 6 = 0.

Solution:

The standard form of the quadratic equation is x² – (sum of the roots) + (product of the roots) = 0
Substitute the values in the equation
Sum of the roots = -b/a = -(-5)/1 = 5/1 = 5
Product of the roots = c/a = 6/1 = 6
Thus the sum and product of the roots of the given quadratic equation are 5 and 6.


Example 3.
If the Quadratic Equation is x² + 9x + 20 what will be the sum of the Roots and product of the Roots?

Solution:

The standard form of the quadratic equation is x² – (sum of the roots) + (product of the roots) = 0
Substitute the values in the equation
Sum of the roots = -b/a = -(9)/1 = -9/1 = -9
Product of the roots = c/a = 20/1 = 20
Thus the sum and product of the roots of the given quadratic equation are -9 and 20.


Example 4.
Determine the discriminant and nature of roots of the quadratic equation x² – 4x + 3 = 0.

Solution:

D = b² – 4ac
a = 1, b = -4, c = 3
= (-4)² – 4(1)(3)
= 16 – 12
= 4
D = 4
Thus the roots of the equation are real, rational, and unequal


Example 5.
Find the discriminant and nature of roots of the quadratic equation x² – 7x – 4 = 0.

Solution:

D = b² – 4ac
a = 1, b = -7, c = -4
= (-7)² – 4(1)(-4)
= 49 + 16
= 65
D = 65
Thus the roots of the equation are real, irrational, and unequal


Example 6.
Find the nature of the roots using the discriminant x² + 4x + 4 = 0

Solution:

We can find the nature of the roots by using the discriminant formula.
D = b² – 4ac
a = 1, b = 4, c = 4
= (4)² – 4(1)(4)
= 16 – 16
D = 0
Thus the roots of the equation are real, rational, and equal


Example 7.
Find the nature of the roots using the discriminant 2x² + 3x + 5 = 0

Solution:

We can find the nature of the roots by using the discriminant formula.
D = b² – 4ac
a = 2, b = 3, c = 5
= (3)² – 4(2)(5)
= 9 – 40
D = -31
Thus the roots of the equation are non-real, imaginary


Example 8.
Find the nature of the roots using the discriminant 3k² + 7k + 8 = 0

Solution:

We can find the nature of the roots by using the discriminant formula.
D = b² – 4ac
a = 3, b = 7, c = 8
= (7)² – 4(3)(8)
= 49 – 96
D = -47
Thus the roots of the equation are non-real, imaginary


Example 9.
Using the discriminant determines the nature of roots of the quadratic equation 9m² – 6m + 1 = 0

Solution:

We can find the nature of the roots by using the discriminant formula.
D = b² – 4ac
a = 9, b = -6, c = 1
= (-6)² – 4(9)(1)
= 36 – 36
D = 0
Thus the roots of the equation are real, rational, and equal.


Example 10.
Form a quadratic equation with real coefficients when one of its roots is (5 – 2i).

Solution:

We know that the imaginary roots always occur in pairs, so the other root is 5 + 2i.
Thus, by getting the sum and product of the roots, we can form the needed quadratic equation.
The sum of the roots = (5 + 2i) + (5 – 2i) = 10
The product of the roots = (5 + 2i)(5 – 2i) = 25 + 4 = 29
The standard form of the quadratic equation is x² – (sum of the roots) + (product of the roots) = 0
x² – 10x + 29 = 0 is the required quadratic equation.


Leave a Comment