If $$\sin(\alpha)$$ and $$\cos(\alpha)$$ are the roots of the equation $$ax^{2}+bx+c=0$$, then $$b^{2}$$ is

$$\sin(\alpha)$$ and $$\cos(\alpha)$$ are the roots of the equation $$ax^{2}+bx+c=0$$. Therefore -

$$\sin\alpha\cos\alpha=\dfrac{c}{a}$$ and

$$\sin\alpha+\cos\alpha=\dfrac{-b}{a}$$

Squaring both sides -

$$\sin^2\alpha+\cos^2\alpha+2\sin\alpha\cos\alpha=\dfrac{b^2}{a^2}$$

$$1+\dfrac{2c}{a}=\dfrac{b^2}{a^2}$$

$$b^2=a^2\left(1+\dfrac{2c}{a}\right)$$

$$b^2=a^2+2ac$$

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