If the arithmetic mean and harmonic mean of t | vedclass

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(A) Let the roots of the quadratic equation be $\alpha$ and $\beta$.Given that the arithmetic mean $AM = \frac{\alpha + \beta}{2} = \frac{3}{2}$,which

Vedclass · Accepted Answer

$x^2 - 3x + 2 = 0$

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(A) Let the roots of the quadratic equation be $\alpha$ and $\beta$.Given that the arithmetic mean $AM = \frac{\alpha + \beta}{2} = \frac{3}{2}$,which implies $\alpha + \beta = 3$.Given that the harmonic mean $HM = \frac{2\alpha\beta}{\alpha + \beta} = \frac{4}{3}$.Substituting $\alpha + \beta = 3$ into the $HM$ formula: $\frac{2\alpha\beta}{3} = \frac{4}{3}$,which simplifies to $2\alpha\beta = 4$,so $\alpha\beta = 2$.The quadratic equation is given by $x^2 - (\alpha + \beta)x + \alpha\beta = 0$.Substituting the values,we get $x^2 - 3x + 2 = 0$.

List-$I$	List-$II$
$(i) \alpha = \beta$	$(A) (ac^2)^{1/3} + (a^2c)^{1/3} + b = 0$
$(ii) \alpha = 2\beta$	$(B) 2b^2 = 9ac$
$(iii) \alpha = 3\beta$	$(C) b^2 = 6ac$
$(iv) \alpha = \beta^2$	$(D) 3b^2 = 16ac$
	$(E) b^2 = 4ac$
	$(F) (ac^2)^{1/3} + (a^2c)^{1/3} = b$

If the arithmetic mean and harmonic mean of the roots of a quadratic equation are $3/2$ and $4/3$ respectively,then what is the equation?

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