Equation with Roots in Terms of Given Roots

Equation with Roots in Terms of Given Roots

## Formula

If $$\alpha,\beta$$ are roots of:

$$ax^2+bx+c=0$$

then the equation whose roots are $$\frac1\alpha,\frac1\beta$$ is:

$$cx^2+bx+a=0$$

The equation whose roots are $$-\alpha,-\beta$$ is:

$$ax^2-bx+c=0$$

The equation whose roots are $$k\alpha,k\beta$$ is:

$$x^2-k(\alpha+\beta)x+k^2\alpha\beta=0$$

## Usage

- Used to construct transformed-root equations quickly.