Real roots of equation `x^2 + 5 |x| + 4 = 0` are

Options

A. `-1, -4`
B. 1, 4
C. `-4, 4`
D. none of these

Answer

To solve the equation \( x^2 + 5|x| + 4 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^2 + 5|x| + 4 = 0 \]
Since \( |x| \) can take two forms (either \( x \) or \( -x \)), we will consider two cases: \( x \geq 0 \) and \( x < 0 \).

### Step 2: Case 1: \( x \geq 0 \)
In this case, \( |x| = x \). Therefore, the equation becomes:
\[ x^2 + 5x + 4 = 0 \]
This is a standard quadratic equation.

### Step 3: Factor the quadratic equation
To factor \( x^2 + 5x + 4 \), we look for two numbers that multiply to \( 4 \) and add up to \( 5 \). The numbers \( 4 \) and \( 1 \) work:
\[ (x + 4)(x + 1) = 0 \]
Setting each factor to zero gives us:
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \quad (\text{not valid since } x \geq 0) \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \quad (\text{not valid since } x \geq 0) \]

### Step 4: Case 2: \( x < 0 \)
In this case, \( |x| = -x \). Therefore, the equation becomes:
\[ x^2 - 5x + 4 = 0 \]

### Step 5: Factor the quadratic equation
To factor \( x^2 - 5x + 4 \), we look for two numbers that multiply to \( 4 \) and add up to \( -5 \). The numbers \( -4 \) and \( -1 \) work:
\[ (x - 4)(x - 1) = 0 \]
Setting each factor to zero gives us:
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \quad (\text{not valid since } x < 0) \]
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \quad (\text{not valid since } x < 0) \]

### Step 6: Conclusion
In both cases, we find that there are no valid solutions for \( x \). Therefore, the equation \( x^2 + 5|x| + 4 = 0 \) has no real roots.

### Final Answer:
The real roots of the equation are **none**.

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