Real roots of the 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 \( x^2 + 5|x| + 4 = 0 \) for its real roots, we will analyze the equation by considering the two cases for the absolute value function \( |x| \).

### Step 1: Split the equation based on the absolute value
The absolute value function \( |x| \) can be expressed as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

### Step 2: Case 1: \( x \geq 0 \)
If \( x \geq 0 \), then \( |x| = x \). The equation becomes:
\[ x^2 + 5x + 4 = 0 \]
Now we will use the quadratic formula to find the roots:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 5 \), and \( c = 4 \).

### Step 3: Calculate the discriminant
First, we calculate the discriminant:
\[ b^2 - 4ac = 5^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9 \]
Since the discriminant is positive, we will have two real roots.

### Step 4: Find the roots
Now substituting back into the quadratic formula:
\[ x = \frac{-5 \pm \sqrt{9}}{2 \cdot 1} = \frac{-5 \pm 3}{2} \]
Calculating the two possible values:
1. \( x = \frac{-5 + 3}{2} = \frac{-2}{2} = -1 \)
2. \( x = \frac{-5 - 3}{2} = \frac{-8}{2} = -4 \)

### Step 5: Check the validity of the roots
Since we assumed \( x \geq 0 \) for this case, we discard both \( x = -1 \) and \( x = -4 \) as they do not satisfy \( x \geq 0 \).

### Step 6: Case 2: \( x < 0 \)
Now we consider the case where \( x < 0 \). Here, \( |x| = -x \). The equation becomes:
\[ x^2 - 5x + 4 = 0 \]
Again, we will use the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} \]

### Step 7: Calculate the discriminant for this case
Calculating the discriminant:
\[ (-5)^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9 \]
Again, the discriminant is positive, indicating two real roots.

### Step 8: Find the roots
Substituting back into the quadratic formula:
\[ x = \frac{5 \pm \sqrt{9}}{2} = \frac{5 \pm 3}{2} \]
Calculating the two possible values:
1. \( x = \frac{5 + 3}{2} = \frac{8}{2} = 4 \)
2. \( x = \frac{5 - 3}{2} = \frac{2}{2} = 1 \)

### Step 9: Check the validity of the roots
Since we assumed \( x < 0 \) for this case, we discard both \( x = 4 \) and \( x = 1 \) as they do not satisfy \( x < 0 \).

### Conclusion
Since both cases yield no valid real roots, we conclude that the equation \( x^2 + 5|x| + 4 = 0 \) has no real roots.

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