A Map of Equations

Try this

1. Click “Fractional eq.”, then “Radical eq.”. Where do their routes join? Apart from the move, what extra line is written on their arrows?
2. Click “Higher degree” and “Quad. system”. Which stop comes next for each?
3. On the whole map, how many routes reach the terminus without passing through “Quadratic eq.”?
4. Click the small stops in dashed boxes (“Complete sq.”, “Discriminant”, “Vieta”, “Integer roots”…). They are not on the main routes. Whom does each of them help?
5. Under every card there is a button to the matching chapter. Which chapter have you not visited yet?

This volume covered many kinds of equations, but only one method, called reduction: if you cannot solve it, turn it into something you can solve.

• unknown in a denominator $\stackrel{\text{clear denominators}}{\to }$ polynomial equation
• unknown under a square root $\stackrel{\text{square}}{\to }$ rational equation
• high degree $\stackrel{\text{reduce the degree}}{\to }$ low degree
• many unknowns $\stackrel{\text{eliminate}}{\to }$ fewer unknowns

Why

Follow one problem along the whole route. With every step, the equation changes into a simpler kind.

Three moves take you down: clearing denominators or squaring, reducing the degree, and eliminating. One habit brings you back: checking.

Remove one condition

Practice

Four problems, four kinds. Find each one on the map first, then get to work.