Solving the Quadratic Inequality

Oct 13, 2025 by wbrum.org 79 views

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Quadratic inequalities are a fundamental concept in algebra and mathematics, and solving them can be a challenging task for many students. In this article, we will focus on solving the quadratic inequality $x^2 + 4x - 21 < 0$ and provide a step-by-step guide on how to solve it.

What is a Quadratic Inequality?

A quadratic inequality is an inequality that involves a quadratic expression, which is a polynomial of degree two. The general form of a quadratic inequality is $ax^2 + bx + c < 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

The Quadratic Formula

Before we dive into solving the quadratic inequality, let's recall the quadratic formula, which is used to solve quadratic equations. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula can be used to find the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$ .

Solving the Quadratic Inequality

To solve the quadratic inequality $x^2 + 4x - 21 < 0$ , we need to find the values of $x$ that make the inequality true. We can start by factoring the quadratic expression:

$x^2 + 4x - 21 = (x + 7)(x - 3)$

Now, we need to find the values of $x$ that make the inequality $(x + 7)(x - 3) < 0$ true.

Finding the Critical Points

The critical points are the values of $x$ that make the inequality true or false. In this case, the critical points are the values of $x$ that make the expression $(x + 7)(x - 3)$ equal to zero. We can find these values by setting each factor equal to zero and solving for $x$ :

$x + 7 = 0 \Rightarrow x = -7$

$x - 3 = 0 \Rightarrow x = 3$

Testing the Intervals

Now that we have found the critical points, we need to test the intervals between them to see which ones make the inequality true. We can do this by choosing a value from each interval and plugging it into the inequality.

Interval 1: $(-\infty, -7)$

Let's choose $x = -10$ as a test value. Plugging it into the inequality, we get:

$(-10 + 7)(-10 - 3) = (-3)(-13) = 39 > 0$

Since the test value is positive, the inequality is false for this interval.

Interval 2: $(-7, 3)$

Let's choose $x = 0$ as a test value. Plugging it into the inequality, we get:

$(0 + 7)(0 - 3) = (7)(-3) = -21 < 0$

Since the test value is negative, the inequality is true for this interval.

Interval 3: $(3, \infty)$

Let's choose $x = 4$ as a test value. Plugging it into the inequality, we get:

$(4 + 7)(4 - 3) = (11)(1) = 11 > 0$

Since the test value is positive, the inequality is false for this interval.

Writing the Solution in Interval Notation

Based on the results of the test values, we can write the solution to the inequality in interval notation:

$(-7, 3)$

This means that the inequality is true for all values of $x$ between $-7$ and $3$ , but not including $-7$ and $3$ .

Conclusion

Solving quadratic inequalities can be a challenging task, but by following the steps outlined in this article, you can solve them with ease. Remember to factor the quadratic expression, find the critical points, test the intervals, and write the solution in interval notation. With practice and patience, you will become proficient in solving quadratic inequalities and be able to tackle more complex problems.

Frequently Asked Questions

What is a quadratic inequality?
How do I solve a quadratic inequality?
What are the critical points of a quadratic inequality?
How do I test the intervals of a quadratic inequality?
What is the solution to the quadratic inequality $x^2 + 4x - 21 < 0$ ?

References

Q: What is a quadratic inequality?

Q: How do I solve a quadratic inequality?

To solve a quadratic inequality, you need to follow these steps:

Factor the quadratic expression, if possible.
Find the critical points by setting each factor equal to zero and solving for $x$ .
Test the intervals between the critical points by choosing a value from each interval and plugging it into the inequality.
Write the solution in interval notation.

Q: What are the critical points of a quadratic inequality?

The critical points are the values of $x$ that make the inequality true or false. In other words, they are the values of $x$ that make the quadratic expression equal to zero.

Q: How do I test the intervals of a quadratic inequality?

To test the intervals, you need to choose a value from each interval and plug it into the inequality. If the test value is positive, the inequality is false for that interval. If the test value is negative, the inequality is true for that interval.

Q: What is the solution to the quadratic inequality $x^2 + 4x - 21 < 0$ ?

The solution to the quadratic inequality $x^2 + 4x - 21 < 0$ is $(-7, 3)$ .

Q: Can I use the quadratic formula to solve quadratic inequalities?

No, you cannot use the quadratic formula to solve quadratic inequalities. The quadratic formula is used to solve quadratic equations, not inequalities.

Q: Can I use a graphing calculator to solve quadratic inequalities?

Yes, you can use a graphing calculator to solve quadratic inequalities. Graphing calculators can help you visualize the solution to the inequality and make it easier to understand.

Q: What is the difference between a quadratic equation and a quadratic inequality?

A quadratic equation is an equation that involves a quadratic expression, while a quadratic inequality is an inequality that involves a quadratic expression. In other words, a quadratic equation is an equation that is true or false, while a quadratic inequality is an inequality that is true or false.

Q: Can I use the same steps to solve quadratic inequalities as I would to solve quadratic equations?

No, you cannot use the same steps to solve quadratic inequalities as you would to solve quadratic equations. While the steps are similar, there are some key differences between solving quadratic equations and solving quadratic inequalities.

Q: Can I use algebraic methods to solve quadratic inequalities?

Yes, you can use algebraic methods to solve quadratic inequalities. Algebraic methods involve using algebraic techniques, such as factoring and testing intervals, to solve the inequality.

Q: Can I use numerical methods to solve quadratic inequalities?

Yes, you can use numerical methods to solve quadratic inequalities. Numerical methods involve using numerical techniques, such as graphing calculators or computer software, to solve the inequality.

Q: What are some common mistakes to avoid when solving quadratic inequalities?

Some common mistakes to avoid when solving quadratic inequalities include:

Not factoring the quadratic expression correctly
Not finding the critical points correctly
Not testing the intervals correctly
Not writing the solution in interval notation correctly

Q: How can I practice solving quadratic inequalities?

You can practice solving quadratic inequalities by working through examples and exercises in your textbook or online resources. You can also try solving quadratic inequalities on your own and checking your answers with a graphing calculator or computer software.

Q: What are some real-world applications of quadratic inequalities?

Quadratic inequalities have many real-world applications, including:

Modeling population growth and decline
Modeling the motion of objects under the influence of gravity
Modeling the spread of diseases
Modeling the behavior of electrical circuits

By understanding how to solve quadratic inequalities, you can apply these concepts to real-world problems and make informed decisions.

Frequently Asked Questions

What is a quadratic inequality?
How do I solve a quadratic inequality?
What are the critical points of a quadratic inequality?
How do I test the intervals of a quadratic inequality?
What is the solution to the quadratic inequality $x^2 + 4x - 21 < 0$ ?