Question

\\3(x - 5)^2 + 2 = -16\\

Explanation:

Isolate the squared term

We begin by subtracting \(2\) from both sides of the equation to isolate the term containing the square.

$$ 3(x - 5)^2 = -16 - 2 $$

$$ 3(x - 5)^2 = -18 $$

Divide by the coefficient

Next, we divide both sides of the equation by \(3\) to fully isolate the squared binomial.

$$ (x - 5)^2 = \frac{-18}{3} $$

$$ (x - 5)^2 = -6 $$

Take the square root

To solve for \(x\), we take the square root of both sides. Since the right side is negative, this introduces the imaginary unit \(i\), where \(i = \sqrt{-1}\).

$$ x - 5 = \pm\sqrt{-6} $$

$$ x - 5 = \pm i\sqrt{6} $$

Solve for x

Finally, we add \(5\) to both sides of the equation to find the complex solutions for \(x\).

$$ x = 5 \pm i\sqrt{6} $$

Answer:

\(x = 5 \pm i\sqrt{6}\)