Question

Solve the following quadratic equation using the perfect square trinomial pattern: 1/4 x^2 + 2x + 4 = 0

Answer 1

To solve the quadratic equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) using the perfect square trinomial pattern, we first want to express it in a form that allows us to factor it as a perfect square.

1. Start by eliminating the fraction. We can multiply the entire equation by 4 to make calculations easier:

\[
4 \left( \frac{1}{4}x^2 + 2x + 4 \right) = 4 \cdot 0
\]
\[
x^2 + 8x + 16 = 0
\]

2. Now the equation is \( x^2 + 8x + 16 = 0 \). Notice that \( x^2 + 8x + 16 \) is a perfect square trinomial. It can be factored as:

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

3. Now, solve for \( x \) by taking the square root of both sides:

\[
x + 4 = 0
\]

4. Subtract 4 from both sides:

\[
x = -4
\]

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

\[
\boxed{-4}
\]