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The Discriminant

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The Discriminant

Study Guide

Key Definition

The discriminant is given by the expression $b^2 - 4ac$ within the quadratic formula, determining the nature of the roots of the quadratic equation.

Important Notes

If $b^2 - 4ac$ is positive, there are two real solutions.
If $b^2 - 4ac$ is zero, there is one real solution.
If $b^2 - 4ac$ is negative, there are two complex solutions.
The discriminant helps in understanding the nature of the roots.
Always check the discriminant before solving the quadratic equation.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

The discriminant $b^2 - 4ac$ tells us about the nature of the roots of a quadratic equation, indicating whether roots are real or complex.

Remember

The discriminant is part of the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Quick Reference

Discriminant Formula:$b^2 - 4ac$

Understanding The Discriminant

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The discriminant is simply $b^2 - 4ac$. It's a part of the quadratic formula.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the discriminant of the equation $x^2 + 4x + 4 = 0$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboarder jumps off a ramp. The height is modeled by $h(t) = -16t^2 + vt + h_0$, where $v = 16 ft/s$ and $h_0 = 5 ft$. Calculate the discriminant to determine if they land smoothly.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Can you find the nature of roots for $x^2 - 6x + 10 = 0$?

Challenge Quiz

Single Choice Quiz

Advanced

For which quadratic equation is the discriminant negative?

Recap

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Review key concepts and takeaways