Questions: Solving Quadratic Equations by Select the correct answer. Solve the following equation by completing the square. 1/4 x^2 + x + 1/4 = 0 A. x = -2 + sqrt(3) or x = -2 - sqrt(3) B. x = -2 + sqrt(3) or x = 2 - sqrt(3) C. x = 1 or x = -5 D. x = 2 + sqrt(3) or x = 2 - sqrt(3)

Solution Steps

We start with the equation

\[ \frac{1}{4} x^{2} + x + \frac{1}{4} = 0 \]

To eliminate the fraction, we multiply the entire equation by 4:

\[ x^{2} + 4x + 1 = 0 \]

Next, we move the constant term to the right side of the equation:

\[ x^{2} + 4x = -1 \]

To complete the square, we add \(\left(\frac{b}{2}\right)^{2}\) to both sides, where \(b = 4\):

\[ x^{2} + 4x + 4 = 3 \]

This simplifies to:

\[ (x + 2)^{2} = 3 \]

Taking the square root of both sides gives us:

\[ x + 2 = \pm \sqrt{3} \]

Thus, we can solve for \(x\):

\[ x = -2 + \sqrt{3} \quad \text{or} \quad x = -2 - \sqrt{3} \]

Final Answer