If the expression ` x^(2) + x + 1 ` is written in the form `( x + (1)/( 2))^(2) + q^(2)` then the possible values of q are

Options

A. `pm (1)/(3)`
B. `pm ( sqrt(3))/(2)`
C. `pm(2)/( sqrt(3))`
D. `pm (1)/(2)`

Answer

To rewrite the expression \( x^2 + x + 1 \) in the form \( \left( x + \frac{1}{2} \right)^2 + q^2 \), we will follow these steps:

### Step 1: Start with the original expression
We have the expression:
\[ x^2 + x + 1 \]

### Step 2: Rewrite the expression
We want to express \( x^2 + x + 1 \) in the form \( \left( x + \frac{1}{2} \right)^2 + q^2 \).

### Step 3: Expand the square
First, we expand \( \left( x + \frac{1}{2} \right)^2 \):
\[ \left( x + \frac{1}{2} \right)^2 = x^2 + 2 \cdot x \cdot \frac{1}{2} + \left( \frac{1}{2} \right)^2 = x^2 + x + \frac{1}{4} \]

### Step 4: Set up the equation
Now, we can set up the equation:
\[ x^2 + x + 1 = \left( x + \frac{1}{2} \right)^2 + q^2 \]
Substituting the expanded form:
\[ x^2 + x + 1 = \left( x^2 + x + \frac{1}{4} \right) + q^2 \]

### Step 5: Simplify the equation
Now, we simplify the equation:
\[ x^2 + x + 1 = x^2 + x + \frac{1}{4} + q^2 \]
Subtract \( x^2 + x \) from both sides:
\[ 1 = \frac{1}{4} + q^2 \]

### Step 6: Isolate \( q^2 \)
Now, isolate \( q^2 \):
\[ q^2 = 1 - \frac{1}{4} \]
\[ q^2 = \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \]

### Step 7: Solve for \( q \)
Taking the square root of both sides gives us:
\[ q = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \]

### Conclusion
Thus, the possible values of \( q \) are:
\[ q = \frac{\sqrt{3}}{2} \quad \text{and} \quad q = -\frac{\sqrt{3}}{2} \]

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