Question
Solve the following quadratic equation: 1/4x^2+2x+4=0
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Answered by
GPT-4o mini
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:
\[
4 \left( \frac{1}{4}x^2 \right) + 4(2x) + 4(4) = 0
\]
This simplifies to:
\[
x^2 + 8x + 16 = 0
\]
Next, we can factor the quadratic equation. We are looking for two numbers that multiply to 16 and add to 8. The numbers 4 and 4 fit this requirement. Thus, we can factor the equation as follows:
\[
(x + 4)(x + 4) = 0
\]
This can also be written as:
\[
(x + 4)^2 = 0
\]
To find the solution, we set the factor equal to zero:
\[
x + 4 = 0
\]
Solving for \(x\), we get:
\[
x = -4
\]
Since the equation \( (x + 4)^2 = 0 \) has only one unique solution (a double root), the solution to the original quadratic equation is:
\[
\boxed{-4}
\]
\[
4 \left( \frac{1}{4}x^2 \right) + 4(2x) + 4(4) = 0
\]
This simplifies to:
\[
x^2 + 8x + 16 = 0
\]
Next, we can factor the quadratic equation. We are looking for two numbers that multiply to 16 and add to 8. The numbers 4 and 4 fit this requirement. Thus, we can factor the equation as follows:
\[
(x + 4)(x + 4) = 0
\]
This can also be written as:
\[
(x + 4)^2 = 0
\]
To find the solution, we set the factor equal to zero:
\[
x + 4 = 0
\]
Solving for \(x\), we get:
\[
x = -4
\]
Since the equation \( (x + 4)^2 = 0 \) has only one unique solution (a double root), the solution to the original quadratic equation is:
\[
\boxed{-4}
\]
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