DISCRIMINANT

Discriminant

Definition Of Discriminant

The Discriminant of an equation gives an idea of the number of roots and the nature of roots of the equation.
If ax² + bx + c = 0 is a quadratic equation, then the Discriminant of the equation, i.e. D = b² - 4ac.

More About Discriminant

If discriminant (D) is equal to 0 then the equation has one real solution.
If D > 0, then the equation has two real solutions.
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Example of Discriminant

The nature of roots of equation 6x² + 11x - 2 = 0 can be found by using discriminant D = b² - 4ac.6x 2 + 11x - 2 = 0
D = b² - 4ac = (11)² - 4(6)(2) [Substitute the values.]
D = 121 - 48 = 73 > 0
As D > 0, the given equation has 2 real solutions.

Video Examples: Free Math Lessons The Discriminant

Solved Example on Discriminant

Ques: Find out the number of solutions the given equation has, by using its discriminant. Check whether the solutions are real or imaginary.
36x 2 + 132x + 121 = 0

Choices:

A. 1 real and 1 imaginary solution
B. 2 real solutions
C. 2 imaginary solutions
D. None of the above
Correct Answer: D

Solution:

Step 1: 36x² + 132x + 121 = 0
Step 2: Compare the equation with the standard form ax² + bx + c = 0 to get the values of a, b and c.
Step 3: b² - 4ac = (132)² - 4(36)(121) [Substitute the values.]
Step 4: = 17424 - 17424 = 0 [Simplify.]
Step 5: Since the discriminant is zero, the quadratic equation has one real solution.

Quick Summary

If D > 0, the equation has two distinct real roots.
If D = 0, the equation has one real root (a repeated root).
If D < 0, the equation has two complex (imaginary) roots.

\[ D = b^2 - 4ac \]

🍎 Teacher Insights

Emphasize the connection between the discriminant's value and the graph of the quadratic function. A positive discriminant corresponds to the graph intersecting the x-axis at two points, a zero discriminant to the graph touching the x-axis at one point, and a negative discriminant to the graph not intersecting the x-axis.

🎓 Prerequisites

Basic Algebra
Solving Equations
Quadratic Formula

Check Your Knowledge

Q1: What is the discriminant of the equation x² + 4x + 4 = 0?

A. 0 B. 4 C. -4 D. 16

Q2: How many real solutions does the equation x² + x + 1 = 0 have?

A. 0 B. 1 C. 2 D. 3

Frequently Asked Questions

Q: What does a negative discriminant mean?
A: A negative discriminant (D < 0) indicates that the quadratic equation has two complex (imaginary) roots, meaning the solutions involve the imaginary unit 'i'.

Q: Can the discriminant be used for equations other than quadratic equations?
A: The discriminant, as defined by b² - 4ac, is specifically for quadratic equations in the standard form ax² + bx + c = 0. Other types of equations may have different methods for determining the nature of their solutions.