Forming Equations with Transformed Roots
Forming Equations with Transformed Roots
A common task gives a quadratic with roots \(\alpha\) and \(\beta\), then asks for a new equation whose roots are a transformation of them — such as \(2\alpha\) and \(2\beta\), or \(\alpha^2\) and \(\beta^2\). You never need the roots themselves: find the new sum and the new product, then write \(x^2-(\text{new sum})\,x+(\text{new product})=0\).
The three-step method
- Find \(\alpha+\beta\) and \(\alpha\beta\) from the given equation.
- Express the new sum and new product in terms of \(\alpha+\beta\) and \(\alpha\beta\), and evaluate them.
- Substitute into \(x^2-(\text{new sum})\,x+(\text{new product})=0\) and clear any fractions.
Common transformations
| New roots | New sum | New product |
|---|---|---|
| \(k\alpha,\ k\beta\) | \(k(\alpha+\beta)\) | \(k^2\alpha\beta\) |
| \(\alpha+k,\ \beta+k\) | \((\alpha+\beta)+2k\) | \(\alpha\beta+k(\alpha+\beta)+k^2\) |
| \(\dfrac{1}{\alpha},\ \dfrac{1}{\beta}\) | \(\dfrac{\alpha+\beta}{\alpha\beta}\) | \(\dfrac{1}{\alpha\beta}\) |
| \(\alpha^2,\ \beta^2\) | \((\alpha+\beta)^2-2\alpha\beta\) | \((\alpha\beta)^2\) |
| \(\alpha^3,\ \beta^3\) | \((\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)\) | \((\alpha\beta)^3\) |
Finish with integer coefficients: multiply through to clear fractions, and remember the "\(=0\)" — an equation, not an expression, is asked for.
The reciprocal-root identities and the displayed ratios require both roots to be nonzero, so \(\alpha\beta\ne0\). Check this condition before using those identities or forming an equation with reciprocal roots.
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