discriminant worksheet

Guri Bee

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22-10-2024

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Mastering the Discriminant: A Comprehensive Worksheet Guide

The discriminant, a key concept in quadratic equations, reveals crucial information about the nature of solutions without actually solving the equation. This article explores the discriminant through a worksheet format, offering explanations and practice exercises to enhance your understanding.

Understanding the Discriminant

What is the discriminant?

The discriminant is a part of the quadratic formula that tells us about the types of solutions a quadratic equation has. It is represented by the symbol Δ and calculated using the formula:

Δ = b² - 4ac

Where:

• a, b, and c are the coefficients of the quadratic equation in the standard form: ax² + bx + c = 0

What does the discriminant tell us?

The discriminant provides insights into the nature of the solutions of a quadratic equation:

• Δ > 0: The quadratic equation has two distinct real roots. This means there are two different solutions, both of which are real numbers.
• Δ = 0: The quadratic equation has one real root (a double root). This means there is only one solution, and it appears twice.
• Δ < 0: The quadratic equation has no real roots. The solutions are complex numbers, involving the imaginary unit "i".

Example:

Let's consider the equation: x² + 4x + 3 = 0

Here, a = 1, b = 4, and c = 3.

Δ = b² - 4ac = 4² - 4 * 1 * 3 = 16 - 12 = 4

Since Δ > 0, the equation has two distinct real roots.

Discriminant Worksheet: Practice Problems

Here are some practice problems to solidify your understanding of the discriminant:

1. Find the discriminant of the following quadratic equations and determine the nature of their solutions:

• a) x² - 6x + 9 = 0
• b) 2x² + 5x - 3 = 0
• c) x² + 2x + 5 = 0

2. For each equation, find the discriminant and then solve the equation using the quadratic formula:

• a) 3x² - 4x + 1 = 0
• b) x² - 8x + 16 = 0
• c) 2x² + 3x + 2 = 0

3. Write a quadratic equation that has:

• a) Two distinct real roots.
• b) One real root (a double root).
• c) No real roots.

Answers:

(1)

• a) Δ = 0, one real root
• b) Δ = 49, two distinct real roots
• c) Δ = -16, no real roots

(2)

• a) Δ = 4, x = 1/3 and x = 1
• b) Δ = 0, x = 4
• c) Δ = -7, x = (-3 ± i√7)/4

(3)

• a) x² - 5x + 6 = 0 (any equation with Δ > 0)
• b) x² - 6x + 9 = 0 (any equation with Δ = 0)
• c) x² + 2x + 5 = 0 (any equation with Δ < 0)

Additional Resources:

• Khan Academy: Discriminant
• PurpleMath: Discriminant

Conclusion:

This worksheet has guided you through the key concepts and practice problems surrounding the discriminant. By understanding the discriminant, you can gain valuable insights into the nature of solutions for any quadratic equation without solving the equation itself. Remember, practice makes perfect! Continuously work through various examples and exercises to master this important concept in algebra.