E_1: a+b+c=0,if 1 is a root of ax^2+bx+c=0. E | vedclass

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(A) Given that $1$ is a root of $ax^2+bx+c=0$.Substituting $x=1$,we get $a(1)^2+b(1)+c=0$,which implies $a+b+c=0$.Therefore,$E_1$ is true.Given that $

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$E_1$ is true,$E_2$ is true

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(A) Given that $1$ is a root of $ax^2+bx+c=0$.Substituting $x=1$,we get $a(1)^2+b(1)+c=0$,which implies $a+b+c=0$.Therefore,$E_1$ is true.Given that $\sin 	heta$ and $\cos 	heta$ are the roots of $ax^2+bx+c=0$.From the properties of roots,we have:$\sin 	heta + \cos 	heta = -\frac{b}{a}$ $(i)$$\sin 	heta \cos 	heta = \frac{c}{a}$ (ii)Squaring equation $(i)$:$(\sin 	heta + \cos 	heta)^2 = \left(-\frac{b}{a}ight)^2$$\sin^2 	heta + \cos^2 	heta + 2 \sin 	heta \cos 	heta = \frac{b^2}{a^2}$Since $\sin^2 	heta + \cos^2 	heta = 1$,we have:$1 + 2\left(\frac{c}{a}ight) = \frac{b^2}{a^2}$Multiplying by $a^2$:$a^2 + 2ac = b^2$$b^2 - a^2 = 2ac$.Therefore,$E_2$ is true.Thus,both $E_1$ and $E_2$ are true.

$E_1: a+b+c=0$,if $1$ is a root of $ax^2+bx+c=0$. $E_2: b^2-a^2=2ac$,if $\sin \theta, \cos \theta$ are the roots of $ax^2+bx+c=0$. Which of the following is true?

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