quadratic inequalities worksheet

Guri Bee

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23-10-2024

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Mastering Quadratic Inequalities: A Comprehensive Worksheet Guide

Quadratic inequalities, like their equation counterparts, involve expressions with a squared term. However, instead of seeking an exact solution, they ask for a range of values that satisfy the inequality. Understanding these inequalities is crucial for various fields, including optimization problems, physics, and economics.

This article uses examples and solutions drawn from a popular Github repository ([link to github repository](link to github repository)) to guide you through solving quadratic inequalities.

Let's begin!

Understanding the Basics

What is a quadratic inequality?

A quadratic inequality is an inequality that involves a quadratic expression. It can be written in the general form:

$ax^2 + bx + c < 0$ or $ax^2 + bx + c > 0$

Where 'a', 'b', and 'c' are real numbers and 'a' is not equal to zero.

How do we solve quadratic inequalities?

1. Factor the quadratic expression: This allows us to identify the roots of the quadratic equation (where the expression equals zero).
2. Create a sign chart: Plot the roots on a number line. This divides the line into intervals. Determine the sign of the expression within each interval.
3. Identify the solution: Based on the inequality symbol (<, >, ≤, ≥), choose the intervals that satisfy the inequality.

Example:

Let's solve the inequality: $x^2 - 4x + 3 < 0$.

1. Factoring: The expression factors as (x - 1)(x - 3). The roots are x = 1 and x = 3.

2. Sign Chart:

     -∞      1      3      +∞
      +     -     + 

• For x < 1, both (x-1) and (x-3) are negative, resulting in a positive product.
• For 1 < x < 3, (x-1) is positive, while (x-3) is negative, giving a negative product.
• For x > 3, both (x-1) and (x-3) are positive, resulting in a positive product.

3. Solution: We want the interval where the expression is negative (< 0). Therefore, the solution is 1 < x < 3.

Graphical Representation:

Another way to visualize the solution is to graph the quadratic function $y = x^2 - 4x + 3$. The solution corresponds to the x-values where the graph lies below the x-axis.

Practical Applications

Quadratic inequalities have various real-world applications:

• Projectile Motion: Calculating the time interval during which a projectile is above a certain height.
• Optimization: Finding the maximum or minimum value of a function within a given constraint.
• Business: Determining the range of production levels that result in profit.

Additional Tips and Resources

• Remember the importance of the inequality sign. This determines whether to include or exclude the roots in the solution.
• Use a graphing calculator or online graphing tools to visually verify your solutions.
• Explore additional examples and exercises from the Github repository to solidify your understanding.

By following the steps outlined in this article, you can confidently solve quadratic inequalities and apply them in various contexts.