Nature of Roots

Discover how the discriminant reveals the nature of quadratic equation solutions!

Explore the Nature of Roots!

Learn how the discriminant determines whether
quadratic equations have real or complex solutions

The Quadratic Formula

The universal solution to any quadratic equation:

x = (-b ± √(b² - 4ac)) / 2a

The expression under the square root (b² - 4ac) is called the discriminant

The discriminant determines the nature of the roots!

Understanding the Discriminant

D = b² - 4ac
If D > 0
Two distinct real roots

The parabola crosses the x-axis at two points

If D = 0
Two equal real roots (one repeated root)

The parabola touches the x-axis at exactly one point

If D < 0
No real roots (complex roots)

The parabola does not cross the x-axis

Interactive Discriminant Calculator

Enter coefficients for ax² + bx + c = 0

a
b
c
Enter coefficients and click Calculate to see the discriminant and nature of roots

Worked Examples

2x² - 4x + 3 = 0
Calculate discriminant:
D = b² - 4ac = (-4)² - 4(2)(3) = 16 - 24 = -8
Since D = -8 < 0, this equation has no real roots
Choose a topic below to explore the nature of quadratic roots!
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📊 Understanding Root Nature
The discriminant is the key to understanding quadratic equation behavior. It tells us not just how many real solutions exist, but also reveals the geometric relationship between the parabola and the x-axis. This fundamental concept connects algebra and geometry beautifully.