Consider a quadratic equation ax^2 + 2bx + c | vedclass

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(C) For the quadratic equation $ax^2 + 2bx + c = 0$ to have no real roots, the discriminant $D < 0$. $D = (2b)^2 - 4ac < 0 \implies 4b^2 < 4ac \implie

Vedclass · Accepted Answer

$a, b, c$ cannot be in $A$.$P$. or $G$.$P$. but can be in $H$.$P$.

Vedclass · Answer

(C) For the quadratic equation $ax^2 + 2bx + c = 0$ to have no real roots, the discriminant $D $D = (2b)^2 - 4ac Since $a, b, c > 0$, we have $b $1$. If $a, b, c$ are in $A$.$P$.,then $2b = a + c$. Since $b $2$. If $a, b, c$ are in $G$.$P$.,then $b^2 = ac$. But we have $b^2 $3$. If $a, b, c$ are in $H$.$P$.,then $b = \frac{2ac}{a+c}$. Since $a, c > 0$, by $A$.$M$. $\geq$ $G$.$M$.,we have $\frac{a+c}{2} \geq \sqrt{ac}$, so $b = \frac{2ac}{a+c} \leq \sqrt{ac}$. The condition $b < \sqrt{ac}$ is satisfied if $a 
eq c$. Thus, $a, b, c$ can be in $H$.$P$.

Consider a quadratic equation $ax^2 + 2bx + c = 0$ where $a, b, c$ are positive real numbers. If the equation has no real root, then which of the following is true?

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