The number of real roots for the eqiuation `x^(2) + 9 | x| + 20 = 0 ` is

Options

A. Zero
B. One
C. Two
D. Three

Explanation

To determine the number of real roots for the equation \( x^2 + 9 |x| + 20 = 0 \), we will analyze the equation by considering the two cases for the absolute value function.

### Step 1: Split the equation based on the absolute value.
The absolute value function \( |x| \) behaves differently based on the sign of \( x \):
- If \( x \geq 0 \), then \( |x| = x \).
- If \( x < 0 \), then \( |x| = -x \).

### Step 2: Case 1 - When \( x \geq 0 \)
In this case, we replace \( |x| \) with \( x \):
\[ x^2 + 9x + 20 = 0 \]
Now, we will factor this quadratic equation.

### Step 3: Factor the quadratic equation
We need two numbers that multiply to \( 20 \) and add up to \( 9 \). The numbers \( 5 \) and \( 4 \) satisfy this condition:
\[ (x + 5)(x + 4) = 0 \]
Setting each factor to zero gives:
\[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \]
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]

### Step 4: Check the validity of the roots
Since we are in the case where \( x \geq 0 \), both roots \( -5 \) and \( -4 \) are not valid because they are negative. Thus, there are no real roots from this case.

### Step 5: Case 2 - When \( x < 0 \)
In this case, we replace \( |x| \) with \( -x \):
\[ x^2 - 9x + 20 = 0 \]
Now, we will factor this quadratic equation.

### Step 6: Factor the quadratic equation
We need two numbers that multiply to \( 20 \) and add up to \( -9 \). The numbers \( -5 \) and \( -4 \) satisfy this condition:
\[ (x - 5)(x - 4) = 0 \]
Setting each factor to zero gives:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]

### Step 7: Check the validity of the roots
Since we are in the case where \( x < 0 \), both roots \( 5 \) and \( 4 \) are not valid because they are positive. Thus, there are no real roots from this case either.

### Conclusion
Since both cases yield no valid real roots, the equation \( x^2 + 9 |x| + 20 = 0 \) has **no real roots**.

### Final Answer
The number of real roots for the equation is **0**.

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