https://file.aiquickdraw.com/mathbot/file/17288293964883dxncdhb.jpg

The equation given in the image is:

\[
3\cos^2(x) + \cos(x) - 4
\]

### Steps to Factor and Simplify:

1. **Substitute \( u = \cos(x) \):**
   This simplifies the equation into a quadratic form:
   \[
   3u^2 + u - 4
   \]
   
2. **Factor the quadratic equation:**
   We need to find factors of \( 3u^2 + u - 4 \). We look for two numbers that multiply to \( 3 \times -4 = -12 \) and add to \( 1 \). The pair \( 4 \) and \( -3 \) works.
   
   Now split the middle term:
   \[
   3u^2 + 4u - 3u - 4
   \]
   Group terms:
   \[
   (3u^2 - 3u) + (4u - 4)
   \]
   Factor each group:
   \[
   3u(u - 1) + 4(u - 1)
   \]
   Factor out the common binomial \( (u - 1) \):
   \[
   (3u + 4)(u - 1)
   \]

3. **Substitute back \( u = \cos(x) \):**
   \[
   (3\cos(x) + 4)(\cos(x) - 1)
   \]

Thus, the factored form of the expression is:
\[
(3\cos(x) + 4)(\cos(x) - 1)
\]

### Final Simplified Answer:
\[
(3\cos(x) + 4)(\cos(x) - 1)
\]

Would you like more details on any part of the process or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you solve a quadratic equation using the factoring method?
2. What are some other trigonometric identities that can be factored like this?
3. How can this factoring method be applied to solve for \( x \) in trigonometric equations?
4. What are alternative methods to solving trigonometric quadratics besides factoring?
5. How does the substitution method simplify complex trigonometric expressions?

**Tip:** When dealing with quadratic trigonometric expressions, always check if substitution can simplify the equation into a solvable form!

Factor and simplify the expression 3cos^2(x) + cos(x) - 4.

Learn how to factor and simplify the trigonometric expression 3cos^2(x) + cos(x) - 4 by converting it into a quadratic equation and applying factoring techniques. A step-by-step guide for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation