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Introduction

Quadratic inequalities are a fundamental concept in algebra, and solving them requires a deep understanding of quadratic equations and their properties. In this article, we will focus on solving the quadratic inequality $f(x) \leq 0$, where $f(x) = x^2 + 2x - 35$. We will use a step-by-step approach to find the solution to this inequality.

Understanding Quadratic Inequalities

A quadratic inequality is an inequality that involves a quadratic expression. In this case, the quadratic expression is $x^2 + 2x - 35$. To solve this inequality, we need to find the values of $x$ that make the expression less than or equal to zero.

Factoring the Quadratic Expression

The first step in solving the quadratic inequality is to factor the quadratic expression. We can factor the expression as follows:

$$x^2 + 2x - 35 = (x + 7)(x - 5)$$

Finding the Critical Points

The critical points of the quadratic expression are the values of $x$ that make the expression equal to zero. In this case, the critical points are $x = -7$ and $x = 5$. These points divide the number line into three intervals: $(-\infty, -7)$, $(-7, 5)$, and $(5, \infty)$.

Testing the Intervals

To determine which intervals satisfy the inequality, we need to test each interval by substituting a test value into the inequality. Let's choose a test value from each interval and substitute it into the inequality.

• For the interval $(-\infty, -7)$, let's choose $x = -10$ as a test value. Substituting $x = -10$ into the inequality, we get:

  $(-10)^2 + 2(-10) - 35 \leq 0$

  Simplifying the expression, we get:

  $100 - 20 - 35 \leq 0$

  $45 \leq 0$

  This is a false statement, so the interval $(-\infty, -7)$ does not satisfy the inequality.

• For the interval $(-7, 5)$, let's choose $x = 0$ as a test value. Substituting $x = 0$ into the inequality, we get:

  $(0)^2 + 2(0) - 35 \leq 0$

  Simplifying the expression, we get:

  $0 + 0 - 35 \leq 0$

  $-35 \leq 0$

  This is a false statement, so the interval $(-7, 5)$ does not satisfy the inequality.

• For the interval $(5, \infty)$, let's choose $x = 10$ as a test value. Substituting $x = 10$ into the inequality, we get:

  $(10)^2 + 2(10) - 35 \leq 0$

  Simplifying the expression, we get:

  $100 + 20 - 35 \leq 0$

  $85 \leq 0$

  This is a false statement, so the interval $(5, \infty)$ does not satisfy the inequality.

Conclusion

Based on the results of the interval testing, we can conclude that the solution to the quadratic inequality $f(x) \leq 0$ is the interval $(-\infty, -7] \cup [5, \infty)$.

Final Answer

Introduction

Quadratic inequalities can be a challenging topic for many students. In this article, we will address some of the most frequently asked questions about quadratic inequalities, including how to solve them, what critical points are, and how to test intervals.

Q: What is a quadratic inequality?

A quadratic inequality is an inequality that involves a quadratic expression. In other words, it is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

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

1. Factor the quadratic expression, if possible.
2. Find the critical points of the quadratic expression by setting the expression equal to zero and solving for $x$.
3. Use the critical points to divide the number line into intervals.
4. Test each interval by substituting a test value into the inequality.
5. Determine which intervals satisfy the inequality.

Q: What are critical points?

Critical points are the values of $x$ that make the quadratic expression equal to zero. In other words, they are the values of $x$ that satisfy the equation $ax^2 + bx + c = 0$. Critical points are important because they divide the number line into intervals, and each interval may satisfy the inequality.

Q: How do I test intervals?

To test an interval, you need to substitute a test value into the inequality. The test value should be a value in the interval that you are testing. If the inequality is true for the test value, then the interval satisfies the inequality. If the inequality is false for the test value, then the interval does not satisfy the inequality.

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

A quadratic equation is an equation that involves a quadratic expression and is equal to zero. For example, $x^2 + 2x - 35 = 0$ is a quadratic equation. A quadratic inequality, on the other hand, is an inequality that involves a quadratic expression and is not equal to zero. For example, $x^2 + 2x - 35 \leq 0$ is a quadratic inequality.

Q: Can I use the quadratic formula to solve a quadratic inequality?

No, you cannot use the quadratic formula to solve a quadratic inequality. The quadratic formula is used to solve quadratic equations, not quadratic inequalities. To solve a quadratic inequality, you need to follow the steps outlined above.

Q: What if I have a quadratic inequality with a negative leading coefficient?

If you have a quadratic inequality with a negative leading coefficient, you need to follow the same steps as above. However, you may need to use the opposite inequality sign. For example, if you have the inequality $-x^2 + 2x - 35 \leq 0$, you need to follow the same steps as above, but use the opposite inequality sign.

Conclusion

Quadratic inequalities can be a challenging topic, but with practice and patience, you can master them. Remember to follow the steps outlined above, and don't be afraid to ask for help if you need it. With this Q&A article, you should have a better understanding of quadratic inequalities and how to solve them.

Final Tips

• Make sure to factor the quadratic expression, if possible.
• Find the critical points of the quadratic expression by setting the expression equal to zero and solving for $x$.
• Use the critical points to divide the number line into intervals.
• Test each interval by substituting a test value into the inequality.
• Determine which intervals satisfy the inequality.

By following these tips and practicing with examples, you should be able to solve quadratic inequalities with ease.