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The references to examples in the textbook are given as well: review as needed before you repeat the self-test. The test could include any of the similar homework problems from these sections: it will NOT just change the numbers in these problems.\\\\
Also, remember that you are graded on your work. I have only indicated what should be done in these problems: you must show complete work to justify your answers in order to receive credit.\\
\textbf{NOTE: Please DO NOT use set notation when reporting your solutions to equations: just write the solutions as I have done below.}
\large
\textbf{Self-Test:} %allow 50 minutes\\\\
For each of problems 1-5: Use the special products patterns (the square of a binomial, difference of squares pattern, or the pattern for the product of a complex number with its conjugate) where appropriate: \textbf{failure to do so will reduce your score.}\\\\
\textbf{Perform the indicated operation: and simplify your answer, using fractions in lowest terms (no decimals). Write the final result in the standard form $a+bi$.}
\begin{description}
\item[1)] $3i(2+5i) = 6i + 15i^{2} = -15 + 6i$
\item[2)] $(3-5i)+(8+6i) = 11 + i$
\item[3)] $(3-5i)(8+6i) = 24 +18i - 40i - 30i^{2} = -6 - 22i$
\item[4)]
$(2-5i)^{2}$ Use the perfect square pattern $(A-B)^{2} = A^{2} - 2AB + B^{2}$: you get $-21 -20i$
\item[5)]
\Large
$\frac{3+i}{3-4i}$ Multiply by the conjugate of the denominator on top and bottom, and use the pattern $(a+bi)(a-bi) = a^{2} + b^{2}$: you get $\frac{1}{5} + \frac{3}{5}i$
\normalsize
\item[6)] \textbf{Solve the equation: make sure to indicate your final answer clearly}\\
\large $2+\sqrt{3x-2}=x$\\
Isolate the radical, then square both sides and solve the resulting quadratic equation using the Zero Product Property. Make sure to check for. ``extraneous solutions''! The final result is $x=6$.
\normalsize
\item[7)] \textbf{Solve the equation using the Square Root Property:}\\
$5x^{2} + 100 = 0$\\
Isolate the $x^{2}$: this gives $x^{2} = -20 \implies x = \pm\sqrt{-20}$\\
Simplify the radical: $x = 2i\sqrt{5}$ or $x= -2i\sqrt{5}$
\item[8)] \textbf{Solve the equation using the Square Root Property:}\\
$(x-7)^{2} = 12 \implies x-7 = \pm\sqrt{12}$\\
Solve for x: $x = 7 + 2\sqrt{3}$ or $x = 7 - \sqrt{3}$
\item[9)] \textbf{Solve the equation by completing the square and using the Square Root Property:}\\
$x^{2} + x + 1 = 0$\\
Move the term +1 to the right-hand side and complete the square:\\
$x^{2} + x + \frac{1}{4} = -1 + \frac{1}{4}$\\
$\left(x + \frac{1}{2}\right)^{2} = -\frac{3}{4}$\\
Use the Square Root Property and solve, you get as final result $x = -\frac{1}{2} + \frac{\sqrt{3}}{2} i$ or $x = -\frac{1}{2} - \frac{\sqrt{3}}{2} i$
\item[10)] \textbf{Solve the equation using the Quadratic Formula:}\\
$2x^{2} - 3x - 1 = 0$\\
You get $x = 1$ or $x = \frac{1}{2}$
\end{description}
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