Question

1. solve the quadratic ( x^2 + 64 = 0 ).

( \bigcirc ) ( x = pm 7 )
( \bigcirc ) ( x = 9 )
( \bigcirc ) ( x = pm 64i )
( \bigcirc ) ( x = pm 8i )

Explanation:

Step 1: Isolate \( x^2 \)

Given the equation \( x^2 + 64 = 0 \), we subtract 64 from both sides to isolate \( x^2 \). So, \( x^2 = - 64 \).

Step 2: Solve for \( x \) using the imaginary unit \( i \)

Recall that \( i^2=-1 \), so we can rewrite \( -64 \) as \( 64\times(-1)=64i^2 \). Then, \( x^2 = 64i^2 \). Taking the square root of both sides, we get \( x=\pm\sqrt{64i^2} \). Since \( \sqrt{64}=8 \) and \( \sqrt{i^2}=|i| = 1 \) (but considering the sign from the square root of a square), we have \( x=\pm 8i \).

Answer:

\( x = \pm 8i \)