Solving the equation z² - 6z + 9 = 4√(z² - 6z - 6)

To solve the given equation, we will follow a step-by-step approach. Let's break down the process into several sections for better understanding.

1. Simplifying the equation:

The equation can be simplified by squaring both sides to eliminate the square root:

(z² - 6z + 9)² = (4√(z² - 6z - 6))²

Simplifying the left side, we get:

(z² - 6z + 9)(z² - 6z + 9) = 16(z² - 6z - 6)

Expanding both sides, we have:

z⁴ - 12z³ + 54z² - 108z + 81 = 16z² - 96z - 96

Combining like terms, we obtain:

z⁴ - 12z³ + 38z² + 12z - 177 = 0

2. Rearranging the equation:

Let's rearrange the equation in descending order of the degree of z:

z⁴ - 12z³ + 38z² + 12z - 177 = 0

3. Factoring the equation:

To factor the equation, we need to find the roots. However, this equation is a quartic equation, which can be quite complex to solve directly. Therefore, we can use numerical or graphical methods to find the approximate roots.

4. Applying numerical methods:

One numerical method to find the roots is the Newton-Raphson method. However, this method requires an initial guess, and it might not always converge to the correct root. Another numerical method is the bisection method, which involves repeatedly bisecting an interval and checking for sign changes. This method guarantees convergence but can be time-consuming.

5. Utilizing graphical methods:

Graphical methods, such as plotting the equation on a graphing calculator or software, can provide an approximate visualization of the roots. By observing the graph, we can estimate the x-values where the function crosses the x-axis, indicating the roots.

6. Conclusion:

In conclusion, the given equation z² - 6z + 9 = 4√(z² - 6z - 6) is a quartic equation that can be quite challenging to solve directly. Therefore, numerical or graphical methods can be used to find the approximate roots. These methods involve iterative calculations, and the accuracy of the solutions depends on the chosen method and initial guess.