Algebra II

What people get wrong

1. Squaring both sides of a radical equation and stopping there — this can introduce extraneous roots. Always substitute each candidate solution back into the original (unsquared) equation.
2. Treating i like an ordinary variable when multiplying complex numbers and forgetting that $i^2=-1$. Simplify every $i^2$ term you produce before combining like terms.
3. Setting up synthetic division with the wrong sign — dividing by $(x-c)$ uses c, but dividing by $(x+c)$ requires using $-c$. Rewrite the divisor in $(x-c)$ form first.
4. Canceling a common factor in a rational expression without noting that the canceled factor's zero is still excluded from the domain. State the restriction before or as you simplify.

What people get wrong

1. Leaving the calculator in the wrong angle mode. If the problem is in radians and the calculator is set to degrees (or vice versa), every trig value comes out wrong even though the keystrokes are correct — always check the mode before computing.
2. Reporting the reference angle as if it were the final answer. The reference angle only gives the magnitude of the trig value; you must still attach the correct sign using the quadrant (ASTC) before the answer is complete.
3. Using the identity $\sin^2\theta+\cos^2\theta=1$ and forgetting the ± choice. Solving for $\sin\theta$ or $\cos\theta$ from the identity gives a square root, which is only correct once you've picked the sign that matches the given quadrant.
4. Confusing amplitude with the maximum value, or midline with zero. Amplitude is half the distance between max and min, and midline is the average of max and min — read both off the graph or context before writing the equation, don't assume midline is always 0.

What people get wrong

1. Writing a domain that matches the algebra but not the real situation (e.g., allowing negative time or negative quantity). Instead, always restrict the domain to values that make sense in context, such as $t\ge0$.
2. Making an arithmetic slip while isolating y when finding an inverse, then not catching it. Instead, verify every inverse by composing $f(f^{-1}(x))$ and confirming it simplifies to x.
3. Treating $\log(a+b)$ as $\log a+\log b$. The addition rule for logs only applies to multiplication inside the log, i.e., $\log(ab)=\log a+\log b$; there is no rule for the log of a sum.
4. Accepting every algebraic solution to a log equation without checking that each log's argument is positive. A root that makes an argument zero or negative must be rejected as extraneous.

What people get wrong

1. Applying the empirical rule to find a percentage at a boundary that isn't exactly 1, 2, or 3 standard deviations from the mean — instead convert to a z-score and use the standard normal table or software for exact boundaries.
2. Concluding that an observational study 'proves' one variable causes another — instead remember that only a randomized controlled experiment supports a causal claim; an observational study can only show association.
3. Treating margin of error as if it fixes bias — instead recognize that MOE only measures variability from random sampling; a biased sampling method (e.g., a self-selected online poll) produces a wrong estimate no matter how small the MOE is.
4. Confusing random sampling with random assignment — instead ask separately 'were subjects chosen randomly from the population?' (affects generalizability) and 'were subjects randomly placed into groups?' (affects causal claims).