Question

solve for d.
$d^2 + 9 = -23$
write your answer in simplified, rationalized form.
$d = \square$ or $d = \square$

Explanation:

Step1: Isolate \(d^2\)

Subtract 9 from both sides of the equation \(d^2 + 9=-23\).
\(d^2=-23 - 9\)
\(d^2=-32\)

Step2: Solve for \(d\)

Take the square root of both sides. Remember that \(\sqrt{-1}=i\).
\(d=\pm\sqrt{-32}\)
Simplify \(\sqrt{-32}\) as \(\sqrt{32}\times\sqrt{-1}\). Since \(\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}\) and \(\sqrt{-1}=i\), we have \(\sqrt{-32}=4i\sqrt{2}\) or \(4\sqrt{2}i\).
So \(d = 4i\sqrt{2}\) or \(d=-4i\sqrt{2}\) (or \(d = 4\sqrt{2}i\) or \(d=-4\sqrt{2}i\))

Answer:

\(d = 4i\sqrt{2}\) or \(d=-4i\sqrt{2}\) (or \(d = 4\sqrt{2}i\) or \(d=-4\sqrt{2}i\))