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Solve the following quadratic equation using the perfect square trinomial pattern: 1/4x^2+2x+4=0
.(1 poin
.(1 poin
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Answered by
GPT-4o mini
To solve the quadratic equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) using the perfect square trinomial pattern, we first want to rearrange the equation so that it can fit the format of a perfect square.
First, we can multiply 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 recognize that \( x^2 + 8x + 16 \) is a perfect square trinomial. This trinomial can be factored as:
\[
(x + 4)^2 = 0
\]
Now, we take the square root of both sides:
\[
x + 4 = 0
\]
Solving for \( x \), we find:
\[
x = -4
\]
Thus, the solution to the quadratic equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) is:
\[
\boxed{-4}
\]
First, we can multiply 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 recognize that \( x^2 + 8x + 16 \) is a perfect square trinomial. This trinomial can be factored as:
\[
(x + 4)^2 = 0
\]
Now, we take the square root of both sides:
\[
x + 4 = 0
\]
Solving for \( x \), we find:
\[
x = -4
\]
Thus, the solution to the quadratic equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) is:
\[
\boxed{-4}
\]