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Factor each quadratic expression that can be factored using integers. Identify those that cannot, and explain why they can't be factored. $$ 2 x^{2}-8 x-10 $$

Many equations cannot be solved directly by backtracking. Some have the variable stated more than once; others involve variable as exponents. Here are some equations that can't be solved directly by backtracking. $$ \begin{array}{c}{f^{2}=f+1} & {x=\sqrt{x}+1 \quad k^{2}+k=0} \\ {1.1^{B}=2} & {\frac{1}{x}=x^{2}+2}\end{array} $$ For each equation below, write yes if it can be solved directly by backtracking and \(n o\) if it cannot. $$ \begin{array}{ll}{\text { a. } 5=\sqrt{x-11}} & {\text { b. } 4^{d}=9} \\\ {\text { c. } 3 g^{2}=5} & {\text { d. } \sqrt{x+1}=x-4}\end{array} $$

Solve each equation by completing the square. $$ 2 x^{2}+4 x-1=0 $$

Factor each quadratic expression that can be factored using integers. Identify those that cannot, and explain why they can't be factored. $$ 3 c^{2}-9 c+6 $$

Solve each equation by factoring using integers, if possible. If an equation can't be solved in this way, explain why. $$ k^{2}+15 k+30=0 $$

Solve each equation by completing the square. $$ 2 u^{2}+3 u-2=0 $$

Write an equation to represent the value of \(B\) in terms of \(r\) $$ \begin{array}{|c|c|}\hline r & {B} \\ \hline 0 & {12} \\ {1} & {4.8} \\ {2} & {1.92} \\ {3} & {0.768} \\ {4} & {0.3072} \\ \hline\end{array} $$

Solve each equation by factoring using integers, if possible. If an equation can't be solved in this way, explain why. $$ n^{2}-17 n+42=0 $$

Stephen, Consuela, and Kwame each made up a number puzzle for their teacher, Mr. Karnowski. 鈥� Stephen said, 鈥淚鈥檓 thinking of a number. If you subtract 1 from my number, square the result, and add 5, you will get 4.鈥� 鈥� Consuela said, 鈥淚鈥檓 thinking of a number. If you subtract 1 from my number, square the result, and add 1, you will get 1.鈥� 鈥� Kwame said, 鈥淚鈥檓 thinking of a number. If you double the number, subtract 5, square the result, and add 1, you will get 10.鈥� After thinking about the puzzles, Mr. Karnowski said, 鈥淥ne of your puzzles has one solution, one of them has two solutions, and one doesn鈥檛 have a solution.鈥� Whose puzzle is which? Write an equation for each puzzle, and explain your answer.

In Exercises 22 and \(23,\) write an equation to represent the situation. Economics the balance \(b\) in a savings account at the end of any year \(t\) if \(\$ 5,000\) is deposited initially and the account earns 8\(\%\) interest per year