Solve each equation. Then, determine if the solutions are rational or irrational.

1. \((x+1)^2 = 4\)
2. \((x-5)^2 = 36\)
3. \((x+3)^2 = 11\)
4. \((x-4)^2 = 6\)

Here is a graph of the equation \(y=81(x-3)^2-4\).

1. According to the graph, what are the solutions to the equation \(81(x-3)^2=4\)?

   

2. Can you tell whether they are rational or irrational? Explain how you know.
3. Solve the equation using a different method and say whether the solutions are rational or irrational. Explain or show your reasoning.

Match each equation to an equivalent equation with a perfect square on one side.

To derive the quadratic formula, we can multiply \(ax^2+bx+c=0\) by an expression so that the coefficient of \(x^2\) is a perfect square and the coefficient of \(x\) is an even number.

1. Which expression, \(a\), \(2a\), or \(4a\), would you multiply \(ax^2+bx+c=0\) by to get started deriving the quadratic formula?
2. What does the equation \(ax^2+bx+c=0\) look like when you multiply both sides by your answer?

Which quadratic expression is in vertex form?

Function \(f\) is defined by the expression \(\frac{5}{x-2}\).

1. Evaluate \(f(12)\).
2. Explain why \(f(2)\) is undefined.
3. Give a possible domain for \(f\).