Roots of `-x^(2) + (1)/(2) x + (1)/(2) = 0` , are

Options

A. `- (1)/(2) , 1`
B. `(1)/(2) , 1`
C. `- (1)/(2) , -1`
D. `(1)/(2) , - (1)/(2)`

Answer

To find the roots of the quadratic equation \(-x^2 + \frac{1}{2}x + \frac{1}{2} = 0\), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ -x^2 + \frac{1}{2}x + \frac{1}{2} = 0 \]

### Step 2: Multiply through by -1
To make the leading coefficient positive, multiply the entire equation by -1:
\[ 2x^2 - x - 1 = 0 \]

### Step 3: Identify coefficients
In the equation \(2x^2 - x - 1 = 0\), identify the coefficients:
- \(a = 2\)
- \(b = -1\)
- \(c = -1\)

### Step 4: Use the quadratic formula
The roots of a quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 5: Substitute the coefficients into the formula
Substituting \(a\), \(b\), and \(c\) into the formula:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]

### Step 6: Simplify the expression
Calculate the discriminant:
\[ (-1)^2 - 4 \cdot 2 \cdot (-1) = 1 + 8 = 9 \]
Now substitute back into the formula:
\[ x = \frac{1 \pm \sqrt{9}}{4} \]

### Step 7: Calculate the square root
Since \(\sqrt{9} = 3\), we have:
\[ x = \frac{1 \pm 3}{4} \]

### Step 8: Find the two roots
Now, calculate the two possible values for \(x\):
1. \(x_1 = \frac{1 + 3}{4} = \frac{4}{4} = 1\)
2. \(x_2 = \frac{1 - 3}{4} = \frac{-2}{4} = -\frac{1}{2}\)

### Final Result
The roots of the equation \(-x^2 + \frac{1}{2}x + \frac{1}{2} = 0\) are:
\[ x_1 = 1 \quad \text{and} \quad x_2 = -\frac{1}{2} \]

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