Questions: Solve (z^4-25 z^2+144=0)

Transcript text: Solve $z^{4}-25 z^{2}+144=0$

Solution

Solution Steps

To solve the equation $ z^4 - 25z^2 + 144 = 0 $, we can use a substitution method. Let $ u = z^2 $, which transforms the equation into a quadratic form: $ u^2 - 25u + 144 = 0 $. We can then solve this quadratic equation for $ u $ using the quadratic formula. Once we find the values of $ u $, we substitute back to find the values of $ z $.

Step 1: Substitute and Transform the Equation

To solve the equation $ z^4 - 25z^2 + 144 = 0 $, we first perform a substitution. Let $ u = z^2 $. This transforms the original equation into a quadratic equation in terms of $ u $:
\[ u^2 - 25u + 144 = 0 \]

Step 2: Solve the Quadratic Equation

We solve the quadratic equation $ u^2 - 25u + 144 = 0 $ using the quadratic formula:
\[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where $ a = 1 $, $ b = -25 $, and $ c = 144 $. Solving this gives us the solutions:
\[ u = 9 \quad \text{and} \quad u = 16 \]

Step 3: Solve for $ z $

Substitute back to find $ z $ using $ z^2 = u $. For each value of $ u $: