Understanding the Equation and Real Roots

The question asks for the number of real roots of the equation \(x^2 + 3|x| + 2 = 0\). A real root is a real number value for \(x\) that satisfies the equation. The equation involves the absolute value function, \(|x|\).

Recall that \(|x|\) is defined as:

• \(|x| = x\) if \(x \ge 0\)
• \(|x| = -x\) if \(x < 0\)

Also, notice that \(x^2 = |x|^2\) for any real number \(x\), since squaring a number always results in a non-negative value, just like the absolute value function. This observation is key to solving this equation efficiently.

Solving the Equation Using Substitution

We can rewrite the equation using the property \(x^2 = |x|^2\):

\(|x|^2 + 3|x| + 2 = 0\)

This looks like a quadratic equation if we consider \(|x|\) as the variable. Let's use a substitution to make this clearer.

Let \(y = |x|\).

Since \(|x|\) is always non-negative for any real number \(x\), our substitution \(y\) must satisfy \(y \ge 0\). The equation becomes:

\(y^2 + 3y + 2 = 0\)

Now we solve this standard quadratic equation for \(y\).

Finding Solutions for y

We can factor the quadratic equation \(y^2 + 3y + 2 = 0\). We need two numbers that multiply to 2 and add up to 3. These numbers are 1 and 2.

So, the equation factors as:

\((y+1)(y+2) = 0\)

This gives us two possible solutions for \(y\):

• \(y+1 = 0 \implies y = -1\)
• \(y+2 = 0 \implies y = -2\)

Checking the Validity of y Solutions

Remember that we made the substitution \(y = |x|\). For \(y\) to be a valid solution in the context of our original problem, it must satisfy the condition \(y \ge 0\) because the absolute value of a real number cannot be negative.

Let's check our solutions for \(y\):

• \(y = -1\): Is \(-1 \ge 0\)? No, \(-1\) is negative.
• \(y = -2\): Is \(-2 \ge 0\)? No, \(-2\) is negative.

Neither of the solutions we found for \(y\) satisfies the necessary condition \(y \ge 0\).

Finding Solutions for x

Since neither value of \(y\) is valid (because they are both negative), there are no real values of \(x\) for which \(|x|\) can equal \(-1\) or \(-2\).

• If \(|x| = -1\), there is no real number \(x\) that satisfies this, as \(|x|\) must be \(\ge 0\).
• If \(|x| = -2\), there is no real number \(x\) that satisfies this, as \(|x|\) must be \(\ge 0\).

Because there are no valid \(y\) values that can equal \(|x|\), there are no real values of \(x\) that satisfy the original equation.

Conclusion on the Number of Real Roots

Based on our analysis, the equation \(x^2 + 3|x| + 2 = 0\) has no real solutions for \(x\). Therefore, it has zero real roots.

Revision Table: Real Roots and Absolute Value Equations

Additional Information: Solving Absolute Value Equations

Equations involving absolute values can be solved using different methods:

• Case Analysis: Split the problem into cases based on the sign of the expression inside the absolute value (e.g., \(x \ge 0\) and \(x < 0\) for \(|x|\)). Solve the equation in each case and check if the solutions fall within the assumed range for that case.
• Substitution: If the equation can be written in terms of \(|f(x)|\) (like our example \(|x|^2 + 3|x| + 2 = 0\)), substitute \(y = |f(x)|\) and solve for \(y\). Then substitute back \(|f(x)| = y\) and solve for \(x\), remembering that \(|f(x)|\) must be non-negative.
• Squaring Both Sides: If an equation is in the form \(|f(x)| = g(x)\), you can square both sides to get \((f(x))^2 = (g(x))^2\). However, this can introduce extraneous roots, so all solutions must be checked in the original equation. For \(|f(x)| = c\) where \(c \ge 0\), the solutions are \(f(x) = c\) or \(f(x) = -c\). If \(c < 0\), there are no solutions.

In the equation \(x^2 + 3|x| + 2 = 0\), notice that \(x^2\) is always non-negative, \(3|x|\) is always non-negative, and \(2\) is positive. The sum of three non-negative terms can only be zero if all terms are zero, but the constant term is 2, which is not zero. Alternatively, the sum of two non-negative terms and a positive term must be positive. \(x^2 \ge 0\), \(3|x| \ge 0\), so \(x^2 + 3|x| + 2 \ge 0 + 0 + 2 = 2\). The left side of the equation is always greater than or equal to 2. Therefore, it can never equal 0. This provides another way to see that there are no real roots.