Solve the Quadratic Equation: x²/4 + x/2 + 1/4 = 0

Solve the following equation:

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

To solve the equation $\frac{x^2}{4}+\frac{x}{2}+\frac{1}{4}=0$, we will follow these steps:

• Convert the equation to standard quadratic form.
• Identify the coefficients $a$, $b$, and $c$.
• Apply the quadratic formula.
• Calculate the discriminant and solve for $x$.

Step 1: The given equation is:
$\frac{x^2}{4} + \frac{x}{2} + \frac{1}{4} = 0$.

To convert it into standard form $ax^2 + bx + c = 0$, multiply the entire equation by 4 to eliminate the denominators:

$x^2 + 2x + 1 = 0$.

Step 2: Identify the coefficients:

• $a = 1$
• $b = 2$
• $c = 1$

Step 3: Apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Step 4: Calculate the discriminant $b^2 - 4ac$:

$b^2 - 4ac = (2)^2 - 4(1)(1) = 4 - 4 = 0$.

Since the discriminant is 0, we have one real repeated solution.

Step 5: Solve for $x$:

$x = \frac{-2 \pm \sqrt{0}}{2(1)} = \frac{-2}{2} = -1$.

Therefore, the solution to the equation is $x = -1$.