Question

Solve the following quadratic equation: 1/4 x^2+2x+4=0

Answer 1

To solve the quadratic equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \), we can start by multiplying the entire equation by 4 to eliminate the fraction:

\[
4 \left( \frac{1}{4}x^2 + 2x + 4 \right) = 0
\]

This simplifies to:

\[
x^2 + 8x + 16 = 0
\]

Next, we can factor the quadratic equation. We look for two numbers that multiply to \( 16 \) (the constant term) and add up to \( 8 \) (the coefficient of \( x \)). The numbers \( 4 \) and \( 4 \) work:

\[
(x + 4)(x + 4) = 0
\]

Thus, we can write:

\[
(x + 4)^2 = 0
\]

Now, we take the square root of both sides:

\[
x + 4 = 0
\]

Solving for \( x \) gives:

\[
x = -4
\]

The solution to the equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) is:

\[
\boxed{-4}
\]

This is a repeated root, meaning that \( x = -4 \) is the only solution with multiplicity 2.