Which of the following conditions implies tha | vedclass

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(C) Given the quadratic equation $\frac{1}{4}x^2 + bx + c = 0$. Multiplying the entire equation by $4$,we get $x^2 + 4bx + 4c = 0$. Using the quadrati

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$b^2 - c$ is the square of an integer and $b$ is an integer

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(C) Given the quadratic equation $\frac{1}{4}x^2 + bx + c = 0$. Multiplying the entire equation by $4$,we get $x^2 + 4bx + 4c = 0$. Using the quadratic formula $x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$ for the equation $x^2 + 4bx + 4c = 0$,where $A=1, B=4b, C=4c$: $x = \frac{-4b \pm \sqrt{(4b)^2 - 4(1)(4c)}}{2(1)}$ $x = \frac{-4b \pm \sqrt{16b^2 - 16c}}{2}$ $x = \frac{-4b \pm 4\sqrt{b^2 - c}}{2}$ $x = -2b \pm 2\sqrt{b^2 - c}$. For $x$ to be an integer,$b$ must be an integer and $b^2 - c$ must be a perfect square of an integer.

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

Which of the following conditions implies that the roots of the equation $\frac{1}{4}x^2 + bx + c = 0$ are integers?

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