[size=150]Find exact solutions (not approximate solutions) to each equation and show your reasoning. Then, say whether you think each solution is rational or irrational. Be prepared to explain your reasoning.[/size][br][br][math]x^2-8=0[/math]

[math](x-5)^2=1[/math]

[math](x-7)^2=2[/math]

[math]\left(\frac{x}{4}\right)^2-5=0[/math]

[table][tr][td][math]2[/math][/td][td][math]3[/math][/td][td][math]\frac{1}{3}[/math][/td][td][math]0[/math][/td][td][math]\sqrt{2}[/math][/td][td][math]\sqrt{3}[/math][/td][td][math]\text{-}\sqrt{3}[/math][/td][td][math]\frac{1}{\sqrt{3}}[/math][/td][/tr][/table][size=150]Here are some statements about the sums and products of numbers. For each statement, decide whether it is [i]always[/i] true, true for [i]some[/i] numbers but not others, or [i]never[/i] true.[br][br]Experiment with sums and products of two numbers in the given list to help you decide.[br][br]Sums:[/size][br][list][*]The sum of two rational numbers is rational.[br][/*][/list]

[list][*]The sum of a rational number and an irrational number is irrational.[br][/*][/list]

[list][*]The sum of two irrational numbers is irrational.[/*][/list]

[size=150]Products:[/size][br][list][*]The product of two rational numbers is rational.[/*][/list]

[list][*]The product of a rational number and an irrational number is irrational.[/*][/list]

[list][*]The product of two irrational numbers is irrational.[br][/*][/list]

[size=150]It can be quite difficult to show that a number is irrational. To do so, we have to explain why the number is impossible to write as a ratio of two integers. It took mathematicians thousands of years before they were finally able to show that [math]\pi[/math] is irrational, and they still don’t know whether or not [math]\pi^{\pi}[/math] is irrational.[br][br]Here is a way we could show that [math]\sqrt{2}[/math] can’t be rational, and is therefore irrational.[br][br][list][*]Let's assume that [math]\sqrt{2}[/math] were rational and could be written as a fraction [math]\frac{a}{b}[/math], where [math]a[/math] and [math]b[/math] are non-zero integers.[/*][*]Let’s also assume that [math]a[/math] and [math]b[/math] are integers that no longer have any common factors. For example, to express 0.4 as [math]\frac{a}{b}[/math], we write [math]\frac{2}{5}[/math] instead of [math]\frac{4}{10}[/math] or [math]\frac{200}{500}[/math]. That is, we assume that [math]a[/math] and [math]b[/math] are 2 and 5, rather than 4 and 10, or 200 and 500.[/*][/list][/size][br]If [math]\sqrt{2}=\frac{a}{b}[/math], then [math]2= \dfrac{\boxed{ }}{\boxed{ }}[/math].

Explain why [math]a^2[/math] must be an even number.

Explain why if [math]a^2[/math] is an even number, then [math]a[/math] itself is also an even number. (If you get stuck, consider squaring a few different integers.)[br]

Because [math]a[/math] is an even number, then [math]a[/math] is 2 times another integer, say, [math]k[/math]. We can write [math]a=2k[/math]. Substitute [math]2k[/math] for [math]a[/math] in the equation you wrote in the first question. Then, solve for [math]b^2[/math].

Explain why the resulting equation shows that [math]b^2[/math], and therefore [math]b[/math], are also even numbers.[br]

We just arrived at the conclusion that [math]a[/math] and [math]b[/math] are even numbers, but given our assumption about [math]a[/math] and [math]b[/math], it is impossible for this to be true. Explain why this is.[br]

Select all solutions that are rational.[br]

[math](x+1)^2=4[/math]

[math](x-5)^2=36[/math]

[math](x+3)^2=11[/math]

[math](x-4)^2=6[/math]

[size=150]Here is a graph of the equation [math]y=81(x-3)^2-4[/math].[/size][br][br][img]data:image/png;base64,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[/img][br]Based on the graph, what are the solutions to the equation [math]81(x-3)^2=4[/math]?

Can you tell whether they are rational or irrational? Explain how you know.[br]

Solve the equation using a different method and say whether the solutions are rational or irrational. Explain or show your reasoning.

[size=150]To derive the quadratic formula, we can multiply [math]ax^2+bx+c=0[/math] by an expression so that the coefficient of [math]x^2[/math] a perfect square and the coefficient of [math]x[/math] an even number.[/size][br][br]Which expression, [math]a[/math], [math]2a[/math], or [math]4a[/math], would you multiply [math]ax^2+bx+c=0[/math] by to get started deriving the quadratic formula?

What does the equation  [math]ax^2+bx+c=0[/math] look like when you multiply both sides by your answer?

[size=150]Which quadratic expression is in vertex form?[/size]

[size=150]Function [math]f[/math] is defined by the expression [math]\frac{5}{x-2}[/math].[/size][br][br]Evaluate [math]f(12)[/math].

Explain why [math]f(2)[/math] is undefined.[br]

Give a possible domain for [math]f[/math].[br]