Questions: Consider the following function. t(x) = (x-6)^2 + 6 Find the x-intercepts, if any. Express the intercept(s) as ordered pair(s).

Solution Steps

To find the \(x\)-intercepts of the function \(t(x) = (x-6)^2 + 6\), we need to determine the values of \(x\) for which \(t(x) = 0\). This involves solving the equation \((x-6)^2 + 6 = 0\). If there are real solutions, they will be the \(x\)-intercepts of the function.

To find the \(x\)-intercepts of the function \(t(x) = (x-6)^2 + 6\), we set the function equal to zero: \[ (x-6)^2 + 6 = 0 \]

Rearranging the equation gives: \[ (x-6)^2 = -6 \] Since the right side is negative, we conclude that there are no real solutions. Instead, we find complex solutions: \[ x - 6 = \pm \sqrt{-6} = \pm i\sqrt{6} \] Thus, the solutions for \(x\) are: \[ x = 6 - i\sqrt{6} \quad \text{and} \quad x = 6 + i\sqrt{6} \]

The \(x\)-intercepts, expressed as ordered pairs, are: \[ (6 - i\sqrt{6}, 0) \quad \text{and} \quad (6 + i\sqrt{6}, 0) \]

Final Answer