Question

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

Answer 1

To solve the quadratic equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) using the perfect square trinomial pattern, we first rearrange the equation.

Let's start by moving the constant to the other side:

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

Next, we want to express the left-hand side as a perfect square trinomial. First, we can factor out \( \frac{1}{4} \) from the left side:

\[
\frac{1}{4}(x^2 + 8x) = -4
\]

Now, we will complete the square for the expression \( x^2 + 8x \). To complete the square, we take half of the coefficient of \( x \) (which is 8), square it, and add it inside the parentheses. Half of 8 is 4, and \( 4^2 = 16 \).

We rewrite the expression:

\[
x^2 + 8x + 16 - 16
\]

Thus,

\[
\frac{1}{4}((x + 4)^2 - 16) = -4
\]

Now simplify the left-hand side:

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

To isolate the square, add 4 to both sides:

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

Now, multiply both sides by 4:

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

Taking the square root of both sides gives:

\[
x + 4 = 0
\]

Solving for \( x \):

\[
x = -4
\]

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

\[
\boxed{-4}
\]