Number of real roots of the equation `x^(2) + 9|x|+20=0` is/are

Options

A. zero
B. one
C. two
D. three

Answer

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

### Step 1: Split the equation based on the absolute value
The absolute value function \( |x| \) can be expressed in two cases:
1. When \( x \geq 0 \), \( |x| = x \)
2. When \( x < 0 \), \( |x| = -x \)

Thus, we can rewrite the original equation as two separate quadratic equations:

1. For \( x \geq 0 \):
\[ x^2 + 9x + 20 = 0 \]

2. For \( x < 0 \):
\[ x^2 - 9x + 20 = 0 \]

### Step 2: Solve the first quadratic equation
Now, we will solve the first equation \( x^2 + 9x + 20 = 0 \) using the quadratic formula:
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = 9 \), and \( c = 20 \):
\[ D = 9^2 - 4 \cdot 1 \cdot 20 = 81 - 80 = 1 \]
Since \( D > 0 \), this equation has 2 real roots.

### Step 3: Solve the second quadratic equation
Next, we solve the second equation \( x^2 - 9x + 20 = 0 \) using the same method:
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = -9 \), and \( c = 20 \):
\[ D = (-9)^2 - 4 \cdot 1 \cdot 20 = 81 - 80 = 1 \]
Again, since \( D > 0 \), this equation also has 2 real roots.

### Step 4: Combine the results
From both cases, we have:
- 2 real roots from the first equation (for \( x \geq 0 \))
- 2 real roots from the second equation (for \( x < 0 \))

Thus, the total number of real roots of the equation \( x^2 + 9|x| + 20 = 0 \) is:
\[ 2 + 2 = 4 \]

### Final Answer
The number of real roots of the equation is **4**.

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