Question
Solve the following quadratic equation using the perfect square trinomial pattern: 1/4x2+2x+4=0
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Answers
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 it into standard form:
1. **Clear the fraction**: Multiply the entire equation by 4 to eliminate the fraction:
\[
4 \left( \frac{1}{4}x^2 + 2x + 4 \right) = 0
\]
This simplifies to:
\[
x^2 + 8x + 16 = 0
\]
2. **Recognize the perfect square trinomial**: The quadratic \( x^2 + 8x + 16 \) can be factored as:
\[
(x + 4)^2 = 0
\]
3. **Solve for x**: To find the values of \( x \), set the perfect square equal to zero:
\[
(x + 4)^2 = 0
\]
Taking the square root of both sides gives:
\[
x + 4 = 0
\]
Thus, solving for \( x \):
\[
x = -4
\]
So, the solution to the quadratic equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) is:
\[
\boxed{-4}
\]
1. **Clear the fraction**: Multiply the entire equation by 4 to eliminate the fraction:
\[
4 \left( \frac{1}{4}x^2 + 2x + 4 \right) = 0
\]
This simplifies to:
\[
x^2 + 8x + 16 = 0
\]
2. **Recognize the perfect square trinomial**: The quadratic \( x^2 + 8x + 16 \) can be factored as:
\[
(x + 4)^2 = 0
\]
3. **Solve for x**: To find the values of \( x \), set the perfect square equal to zero:
\[
(x + 4)^2 = 0
\]
Taking the square root of both sides gives:
\[
x + 4 = 0
\]
Thus, solving for \( x \):
\[
x = -4
\]
So, the solution to the quadratic equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) is:
\[
\boxed{-4}
\]
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