https://file.aiquickdraw.com/mathbot/file/1726096970537079vhqng.jpg

Let's break down and solve the given problems from the image:

**1.1 Giải các phương trình sau (Solve the following equations):**

### a) \( |x^2 + 2| = |x^2 - 1| \)

This equation involves absolute values. To solve it, we need to break it down into different cases based on the conditions where the expressions inside the absolute values change sign.

#### Case 1: \( x^2 + 2 \geq 0 \) and \( x^2 - 1 \geq 0 \)
In this case, both absolute values drop, and we solve the equation:
\[
x^2 + 2 = x^2 - 1
\]
Simplifying:
\[
2 = -1 \quad (\text{which is false}).
\]
So, no solution in this case.

#### Case 2: \( x^2 + 2 \geq 0 \) and \( x^2 - 1 < 0 \)
Here, the equation becomes:
\[
x^2 + 2 = -(x^2 - 1)
\]
Simplifying:
\[
x^2 + 2 = -x^2 + 1
\]
\[
2x^2 = -1
\]
Again, this equation has no solution since \( x^2 \geq 0 \), so \( 2x^2 = -1 \) is impossible.

#### Case 3: \( x^2 + 2 < 0 \)
There is no solution here either, as \( x^2 + 2 \geq 2 \) is always non-negative for all real \( x \).

### b) \( \cos 2x + \cos x + 1 = 0 \) on \( [0, 2\pi] \)

We solve this trigonometric equation by using the identity for \( \cos 2x \):
\[
\cos 2x = 2\cos^2 x - 1
\]
Substituting this into the equation:
\[
2\cos^2 x - 1 + \cos x + 1 = 0
\]
Simplifying:
\[
2\cos^2 x + \cos x = 0
\]
Factoring:
\[
\cos x (2\cos x + 1) = 0
\]
This gives two cases:
1. \( \cos x = 0 \)
2. \( 2\cos x + 1 = 0 \)

#### Case 1: \( \cos x = 0 \)
This happens when:
\[
x = \frac{\pi}{2}, \frac{3\pi}{2} \quad \text{(in the interval } [0, 2\pi])
\]

#### Case 2: \( 2\cos x + 1 = 0 \)
Solving:
\[
\cos x = -\frac{1}{2}
\]
This happens when:
\[
x = \frac{2\pi}{3}, \frac{4\pi}{3}
\]

### Final solution for part (b):
The solutions are:
\[
x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{2\pi}{3}, \frac{4\pi}{3}
\]

---

Do you need further details or have any questions on these solutions?

Here are some related questions to expand on the current problem:

1. What is the general method to solve absolute value equations?
2. How do trigonometric identities help simplify trigonometric equations?
3. Can you generalize how to solve quadratic trigonometric equations like in part (b)?
4. How would the solution change if the interval for \( x \) in part (b) was different?
5. What other types of transformations are useful in solving complex trigonometric equations?

**Tip:** When dealing with absolute values, always consider the different cases where the expressions inside the absolute signs change behavior (i.e., positive or negative).

Solve the following equations: a) |x^2 + 2| = |x^2 - 1|, b) cos 2x + cos x + 1 = 0 on the interval [0, 2π]

Learn how to solve absolute value equations and trigonometric equations, including |x^2 + 2| = |x^2 - 1| and cos 2x + cos x + 1 = 0. This step-by-step guide covers algebraic and trigonometric concepts, including factoring and applying trigonometric identities.

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