Polynomial Having Square Roots of Original Roots

Polynomial Having Square Roots of Original Roots

## Formula

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

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

then:

$$(\sqrt\alpha+\sqrt\beta)^2=\alpha+\beta+2\sqrt{\alpha\beta}$$

$$=( -\frac ba)+2\sqrt{\frac ca}$$

and:

$$(\sqrt\alpha-\sqrt\beta)^2=\alpha+\beta-2\sqrt{\alpha\beta}$$

$$=( -\frac ba)-2\sqrt{\frac ca}$$

## Conditions / Special Cases

These expressions require the relevant square roots to be defined in the chosen number system.

## Usage

- Used to evaluate expressions involving square roots of quadratic roots.