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(A) Given that $\sin \alpha$ and $\cos \alpha$ are the roots of $ax^2 + bx + c = 0$. By the properties of roots,we have: $\sin \alpha + \cos \alpha =

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$a^2 - b^2 + 2ac = 0$

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(A) Given that $\sin \alpha$ and $\cos \alpha$ are the roots of $ax^2 + bx + c = 0$. By the properties of roots,we have: $\sin \alpha + \cos \alpha = -\frac{b}{a}$ $\sin \alpha \cos \alpha = \frac{c}{a}$ We know the identity: $\sin^2 \alpha + \cos^2 \alpha = 1$ This can be written as: $(\sin \alpha + \cos \alpha)^2 - 2\sin \alpha \cos \alpha = 1$ Substituting the values: $(-\frac{b}{a})^2 - 2(\frac{c}{a}) = 1$ $\frac{b^2}{a^2} - \frac{2c}{a} = 1$ Multiplying by $a^2$: $b^2 - 2ac = a^2$ Rearranging the terms: $a^2 - b^2 + 2ac = 0$

If $\sin \alpha$ and $\cos \alpha$ are the roots of the equation $ax^2 + bx + c = 0$,then:

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