Question

What are the roots of g(x)=4x^2+256 ? Provide the exact answers. Enter the smaller number first. (1 point) The roots are □ and □

Answer 1

To find the roots of the function \( g(x) = 4x^2 + 256 \), we can set the equation equal to zero:

\[
4x^2 + 256 = 0
\]

Next, we isolate \( x^2 \) by subtracting 256 from both sides:

\[
4x^2 = -256
\]

Dividing both sides by 4 gives:

\[
x^2 = -64
\]

Taking the square root of both sides, we remember to include both the positive and negative roots:

\[
x = \pm \sqrt{-64}
\]

Since \( \sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i \) (where \( i \) is the imaginary unit), the roots can be expressed as:

\[
x = 8i \quad \text{and} \quad x = -8i
\]

Listing the roots, we have:

\[
-8i \quad \text{and} \quad 8i
\]

Therefore, the exact roots of the function are:

\[
\boxed{-8i} \quad \text{and} \quad \boxed{8i}
\]