Key Concepts and Formulas

• Absolute Value Definition: The absolute value of a real number $x$, denoted by $|x|$, is its distance from zero. $|x| = x$ if $x \ge 0$, and $|x| = -x$ if $x < 0$. Crucially, $|x| \ge 0$ for all real $x$.
• Squaring and Absolute Value: For any real number $x$, $x^2 = |x|^2$. This allows us to rewrite equations involving both $x^2$ and $|x|$.
• Solving Quadratic Equations: A quadratic equation of the form $ax^2 + bx + c = 0$ can be solved by factoring, completing the square, or using the quadratic formula.

Step-by-Step Solution

1. Rewrite the Equation in Terms of |x| The given equation is: $x^2 - |x| - 12 = 0$ We use the property $x^2 = |x|^2$ to rewrite the equation as a quadratic in $|x|$: $|x|^2 - |x| - 12 = 0$

   • Why this step? This substitution transforms the original equation into a standard quadratic equation, making it solvable using familiar techniques.

2. Solve the Quadratic Equation for |x| Let $y = |x|$. The equation becomes: $y^2 - y - 12 = 0$ We factor this quadratic equation. We look for two numbers that multiply to -12 and add up to -1. These numbers are -4 and 3. $(y - 4)(y + 3) = 0$ This gives us two possible values for $y$: $y - 4 = 0 \quad \Rightarrow \quad y = 4$ $y + 3 = 0 \quad \Rightarrow \quad y = -3$

   • Why this step? Factoring simplifies finding the potential values of $|x|$.

3. Analyze the Possible Values for |x| and Solve for x Substituting $|x|$ back for $y$, we have: $|x| = 4 \quad \text{or} \quad |x| = -3$

   • Why this step? We need to determine which values of $|x|$ yield valid solutions for $x$.

   • Case 1: $|x| = -3$ Since the absolute value of a real number cannot be negative, the equation $|x| = -3$ has no real solutions.

     • Why this step? It's crucial to discard extraneous solutions that violate the properties of absolute value.

   • Case 2: $|x| = 4$ This equation means that $x$ is either 4 or -4. $x = 4 \quad \text{or} \quad x = -4$

     • Why this step? Applying the definition of absolute value, $|x| = a$ implies $x = a$ or $x = -a$ when $a>0$.

4. Count the Total Number of Real Solutions The real solutions are $x = 4$ and $x = -4$. Therefore, there are 2 distinct real solutions to the original equation.

Common Mistakes & Tips

• Forgetting the Negative Root: When solving $|x| = k$ (where $k > 0$), remember to consider both $x = k$ and $x = -k$.
• Ignoring the Non-Negativity of Absolute Value: Always remember that $|x| \ge 0$. Discard any solutions that lead to a negative value for $|x|$.
• Not Utilizing $x^2 = |x|^2$: Recognizing and applying this identity simplifies the problem into solving a quadratic equation.

Summary

The equation $x^2 - |x| - 12 = 0$ is solved by recognizing that $x^2 = |x|^2$, transforming the equation into $|x|^2 - |x| - 12 = 0$. Solving for $|x|$ yields $|x| = 4$ (since $|x| = -3$ is not possible). The equation $|x| = 4$ gives two solutions, $x = 4$ and $x = -4$. Therefore, the total number of real solutions is 2.

Final Answer The final answer is \boxed{2}, which corresponds to option (A).