The number of real solutions of the equation `2|x|^2-5|x|+2=0` is:

Options

A. 0
B. 4
C. 2
D. none of these

Answer

To find the number of real solutions of the equation \(2|x|^2 - 5|x| + 2 = 0\), we can follow these steps:

### Step 1: Substitute \(y = |x|\)
Since the equation involves \(|x|\), we can let \(y = |x|\). This transforms our equation into:
\[ 2y^2 - 5y + 2 = 0 \]

### Step 2: Solve the quadratic equation
Next, we will solve the quadratic equation \(2y^2 - 5y + 2 = 0\) using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 2\), \(b = -5\), and \(c = 2\).

### Step 3: Calculate the discriminant
First, we calculate the discriminant \(D\):
\[ D = b^2 - 4ac = (-5)^2 - 4 \cdot 2 \cdot 2 = 25 - 16 = 9 \]

### Step 4: Find the roots
Since the discriminant is positive (\(D > 0\)), we have two distinct real roots. Now we can find the roots:
\[ y = \frac{5 \pm \sqrt{9}}{2 \cdot 2} = \frac{5 \pm 3}{4} \]
Calculating the two roots:
1. \(y_1 = \frac{5 + 3}{4} = \frac{8}{4} = 2\)
2. \(y_2 = \frac{5 - 3}{4} = \frac{2}{4} = \frac{1}{2}\)

### Step 5: Determine the corresponding \(x\) values
Since \(y = |x|\), we have:
1. For \(y_1 = 2\), \(x = 2\) or \(x = -2\)
2. For \(y_2 = \frac{1}{2}\), \(x = \frac{1}{2}\) or \(x = -\frac{1}{2}\)

### Step 6: Count the total number of solutions
Thus, we have four solutions for \(x\):
1. \(x = 2\)
2. \(x = -2\)
3. \(x = \frac{1}{2}\)
4. \(x = -\frac{1}{2}\)

### Conclusion
The total number of real solutions to the equation \(2|x|^2 - 5|x| + 2 = 0\) is **4**.

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