Key Concepts and Formulas

• Absolute Value Property: For any real number $x$, $x^2 = |x|^2$.
• Definition of Absolute Value: $|x| = x$ if $x \geq 0$, and $|x| = -x$ if $x < 0$.
• Solving Quadratic Equations: Factoring, quadratic formula.

Step-by-Step Solution

1. Rewriting the Equation using the Absolute Value Property The given equation is: $x^2 - 3|x| + 2 = 0$ We use the property $x^2 = |x|^2$ to rewrite the equation: $|x|^2 - 3|x| + 2 = 0$ Explanation: This substitution allows us to treat the equation as a quadratic in terms of $|x|$, which simplifies the solving process.

2. Substitution for Clarity Let $y = |x|$. Substituting $y$ into the equation from Step 1, we get: $y^2 - 3y + 2 = 0$ Explanation: This substitution transforms the equation into a standard quadratic form, $ay^2 + by + c = 0$, making it easier to analyze and solve.

3. Solving the Quadratic Equation for y Now we solve the quadratic equation $y^2 - 3y + 2 = 0$ for $y$. We can solve this by factoring. We need two numbers that multiply to $+2$ and add up to $-3$. These numbers are $-1$ and $-2$. Therefore, we can factor the quadratic equation as: $(y - 1)(y - 2) = 0$ Explanation: Factoring allows us to find the roots (solutions) of the quadratic equation by setting each factor equal to zero. This gives us two possible values for $y$: $y - 1 = 0 \implies y = 1$ $y - 2 = 0 \implies y = 2$

4. Substituting Back and Solving for x We have found the possible values for $y$, which represents $|x|$. Now we must substitute back $y = |x|$ and solve for $x$. We consider each value of $y$ separately:

• Case 1: $|x| = 1$ Explanation: This equation states that the absolute value of $x$ is 1. This means $x$ is a number whose distance from zero on the number line is 1. Therefore, $x$ can be either 1 or -1. $x = 1 \quad \text{or} \quad x = -1$

• Case 2: $|x| = 2$ Explanation: This equation states that the absolute value of $x$ is 2. This means $x$ is a number whose distance from zero on the number line is 2. Therefore, $x$ can be either 2 or -2. $x = 2 \quad \text{or} \quad x = -2$

5. Consolidating All Real Solutions By considering all possible cases for $|x|$, we have found all the real values of $x$ that satisfy the original equation. The set of solutions is: $\{1, -1, 2, -2\}$ Explanation: We combine all unique solutions obtained from each case to form the complete solution set for the original equation.

6. Counting the Number of Solutions The set of solutions is $\{1, -1, 2, -2\}$. By counting the distinct elements in this set, we find that there are four distinct real solutions.

Common Mistakes & Tips

• Remember the two solutions for absolute value: When solving $|x| = a$ where $a > 0$, remember that $x$ can be both $a$ and $-a$.
• Check for extraneous solutions: Although not applicable in this problem, always check your solutions in the original equation, especially when dealing with absolute values.
• Non-negative absolute value: Remember that $|x|$ must always be non-negative. If you obtain a negative value for $|x|$, discard that solution.

Summary

The given equation $x^2 - 3|x| + 2 = 0$ was solved by first recognizing that $x^2 = |x|^2$. Substituting $y = |x|$ transformed the equation into a quadratic equation $y^2 - 3y + 2 = 0$. Solving for $y$ yielded $y = 1$ and $y = 2$. Substituting back $|x| = 1$ gave $x = \pm 1$, and $|x| = 2$ gave $x = \pm 2$. Therefore, there are a total of four real solutions.

Final Answer The final answer is $\boxed{4}$, which corresponds to option (C).