Quadratics: discriminant first
Discriminant first, always: D = b² − 4ac tells the whole story before any formula — D > 0 two real roots, D = 0 one repeated root, D < 0 no real roots (complex pair).
Then roots are (−b ± √D) / 2a, and the vertex sits at x = −b/2a. Reading D first turns 'solve it' into 'predict, then confirm' — faster and fewer sign errors.
Mistakes: dividing by a without checking a ≠ 0 (it is linear then), losing the ± (half the solutions vanish), and rounding √D mid-computation instead of at the end.
Example: x² − 5x + 6 = 0 → D = 25 − 24 = 1 → x = (5 ± 1)/2 → x₁ = 3, x₂ = 2. Vertex at 2.5.
Try: Quadratic with Steps · Linear Equations
Derivatives as rates, integrals as totals
Derivatives are rates: how fast is this changing right now. Position → velocity → acceleration is one derivative per arrow. That reading turns symbol-pushing into physics you can feel.
Integrals are totals: area under the rate curve gives back the accumulation. Velocity's area over 10 s is distance traveled — the Fundamental Theorem in one sentence.
Mistakes: differentiating term-by-term but dropping the chain rule (sin(x²) needs ×2x), forgetting +C in indefinite integrals, and mixing average rate with instantaneous rate in word problems.
Example: s(t) = 3t² → v = 6t → at t=5, 30 m/s. Integrate v from 0 to 5: 3·25 = 75 m traveled. Rate in, total out.
Try: Derivative (power/polynomial) · Integral (power rule)
Mean vs median: which to trust
Mean answers 'what is typical if everything pools together'; median answers 'what does the middle case look like'. Skew decides: incomes, house prices, wait times — median resists the billionaires, mean doesn't.
Check the shape first: symmetric → mean is fine and uses all data; skewed or with outliers → median (plus quartiles) tells the honest story. Report both when they disagree — the gap is the insight.
Mistakes: averaging averages of different-sized groups (weight them), quoting mean salary to describe 'normal', and ignoring n (a 60% from n=10 is noise).
Example: [4, 8, 15, 16, 23, 42] → mean 18, median 15.5. Add one billionaire and the mean explodes; the median shrugs.
Try: Mean/Median/Mode/σ · Permutations & Combinations
Graph any function free
Type f(x) and get 160 sampled points plus a line chart: roots, peaks, and asymptotes become visible instead of abstract. Gaps (division by zero) are skipped, never connected — 1/x never draws its infamous wall.
It matters because graphing is where algebra clicks: the discriminant tells you how many roots, the plot shows where they live.
Mistakes: trusting the chart past the sampled range (zoom out before concluding), missing narrow spikes between samples, and degree/radian mode in trig plots.
Example: x^2 on [-2,2] gives a parabola with vertex (0,0); 1/x shows two clean branches with the gap at zero.
Try: Function Plotter f(x) · Quadratic with Steps
Percentages: 3 modes, zero confusion
Three questions share one word: percent-of (200 × 15% = 30), percent change (80 to 100 = +25%), and is-what-percent-of (50 is 25% of 200). Each divides by a different base — that is the entire confusion.
It matters because stores, salaries, and statistics each use a different mode: discounts are percent-of, growth is percent change, shares are is-what-percent-of.
Mistakes: averaging percentages of different bases, confusing percentage points with percent (5% to 7% is +2pp, +40%), and zero bases (change from 0 is undefined, not infinite).
Example: 80 to 100 is +25%; 100 to 80 is −20%. Same pair, different base, different answer — run both modes.
Try: Percentages · Linear Inequalities