If the roots of the quadratic equation ax^2 - | vedclass

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Question

(A) Let the roots of the equation $ax^2 - bx - c = 0$ be $\alpha$ and $\beta$.If the roots are shifted by a constant $\lambda$,the new roots become $\

Vedclass · Accepted Answer

$\frac{b^2 - 4ac}{a^2}$

Vedclass · Answer

(A) Let the roots of the equation $ax^2 - bx - c = 0$ be $\alpha$ and $\beta$.If the roots are shifted by a constant $\lambda$,the new roots become $\alpha + \lambda$ and $\beta + \lambda$.The difference between the roots remains invariant:$|(\alpha + \lambda) - (\beta + \lambda)| = |\alpha - \beta|$.We know that the difference of roots is given by $|\alpha - \beta| = \frac{\sqrt{b^2 - 4ac}}{|a|}$.Squaring both sides,we get $(\alpha - \beta)^2 = \frac{b^2 - 4ac}{a^2}$.Therefore,the expression $\frac{b^2 - 4ac}{a^2}$ remains unchanged.

If the roots of the quadratic equation $ax^2 - bx - c = 0$ are shifted by a constant value,which of the following expressions involving $a, b, c$ remains unchanged?

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