Lens 05 of 9

The easy instance

Look at whether the answer depends only on a ratio, or holds for every input. Then use the simplest values.

If a question says 'for any a, b, c' or gives only a ratio, the answer is the same for every legal input. So it is the same for the easiest one. Pick roots 3 and 4, or P = 1 and Q = 3, compute concretely, and read the general pattern off the result.

The clues that call for it

Polynomials

A question about roots in general, with letters instead of numbers

Let the roots be 3 and 4, build x² − 7x + 12, and compare.

Some Applications of Trigonometry

Only a ratio is given

The shadow is √3 times the height: take the height as 1.

Anywhere

An expression that depends on P/Q only

P/Q = 1/3 → let P = 1 and Q = 3, then compute.

Worked example

α and β are the roots of ax² + bx + c = 0. Find the equation whose roots are 1/α and 1/β.

1
Pick the easiest legal instance
Let the roots be 3 and 4.
x² − 7x + 12 = 0 (a = 1, b = −7, c = 12)
2
Compute the target concretely
The reciprocal roots are 1/3 and 1/4.
12x² − 7x + 1 = 0
3
Pattern-match
a and c have swapped; b is unchanged.
cx² + bx + a = 0
4
Guardrail
Only valid when the target really is input-independent. Confirm with a second set of values before trusting it.

Try the lens

The easy instance

The clues that call for it

α and β are the roots of ax² + bx + c = 0. Which equation has roots −α and −β?

Problems that use this lens