Question
Solve the equation by finding square roots.4a Superscript 2plus64equals0Question content area bottomPart 1Select the correct choice below and, if necessary, fill in the answer box to complete your choice.A.aequalsenter your response here (Simplify your answer. Use a comma to separate answers as needed.)B.The solution is not a real number.
Answers
GPT-4o mini
To solve the equation \(4a^2 + 64 = 0\), we can start by isolating \(a^2\).
1. Subtract 64 from both sides:
\[
4a^2 = -64
\]
2. Divide both sides by 4:
\[
a^2 = -16
\]
Next, to find the values of \(a\), we take the square root of both sides. However, since we are taking the square root of a negative number, we will get complex solutions:
3. Taking the square root:
\[
a = \pm \sqrt{-16}
\]
Using the property that \(\sqrt{-1} = i\), we can rewrite the equation:
\[
a = \pm 4i
\]
Thus, the solutions are:
\[
a = 4i, -4i
\]
So, the correct choice is:
A. \(a = 4i, -4i\) (Simplified answer: \(4i, -4i\))
1. Subtract 64 from both sides:
\[
4a^2 = -64
\]
2. Divide both sides by 4:
\[
a^2 = -16
\]
Next, to find the values of \(a\), we take the square root of both sides. However, since we are taking the square root of a negative number, we will get complex solutions:
3. Taking the square root:
\[
a = \pm \sqrt{-16}
\]
Using the property that \(\sqrt{-1} = i\), we can rewrite the equation:
\[
a = \pm 4i
\]
Thus, the solutions are:
\[
a = 4i, -4i
\]
So, the correct choice is:
A. \(a = 4i, -4i\) (Simplified answer: \(4i, -4i\))