Question

solve for z.
-157z² - 12 = -61z²
write your answer in simplified, rationalized form.
z = or z =
(buttons: i, fraction, square root)

Explanation:

Step1: Add \( 157z^2 \) to both sides

To isolate the \( z^2 \) terms, we add \( 157z^2 \) to each side of the equation \( -157z^2 - 12 = -61z^2 \). This gives us:
\( -12 = -61z^2 + 157z^2 \)
Simplifying the right - hand side, we combine like terms: \( -61z^2+157z^2 = 96z^2 \), so the equation becomes \( - 12=96z^2 \)

Step2: Divide both sides by 96

We want to solve for \( z^2 \), so we divide both sides of the equation \( -12 = 96z^2 \) by 96.
\( z^2=\frac{- 12}{96}\)
Simplify the fraction: \( \frac{-12}{96}=-\frac{1}{8}\)

Step3: Take the square root of both sides

We take the square root of both sides of the equation \( z^2 =-\frac{1}{8}\). Remember that \( \sqrt{-a}=i\sqrt{a}\) for \( a>0 \).
\( z=\pm\sqrt{-\frac{1}{8}}\)
Simplify \( \sqrt{-\frac{1}{8}}\):
First, we can rewrite \( \sqrt{-\frac{1}{8}}\) as \( i\sqrt{\frac{1}{8}}\). Then, we simplify \( \sqrt{\frac{1}{8}}\) by rationalizing the denominator. \( \sqrt{\frac{1}{8}}=\frac{\sqrt{1}}{\sqrt{8}}=\frac{1}{2\sqrt{2}}=\frac{\sqrt{2}}{4}\) (multiplying numerator and denominator by \( \sqrt{2} \) to rationalize the denominator). So \( z = \pm\frac{i\sqrt{2}}{4}\)

Answer:

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