If a-b-c are integers and b2-4ac-5d2- d - N- then roots of the quadratic equation ax2-bx-c-0

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Addition and Subtraction of Surds

If a,b,c ar...

Question

If $a, b, c$ are integers and $b^{2} = 4 (a c + 5 d^{2}), d \in N$ , then roots of the quadratic equation $a x^{2} + b x + c = 0$

Irrational

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Rational & different

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Complex conjugate

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Rational & equal

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Solution

The correct option is C Irrational
$b^{2} = 4 (a c + 5 d^{2})$
$D = b^{2} - 4 a c$
$= 4 (a c + 5 d^{2}) - 4 a c$
$= 4 a c + 20 d^{2} - 4 a c$
$= 20 d^{2}$
Now, $x = \frac{- b \pm \sqrt{D}}{2 a}$
$= \frac{- b \pm \sqrt{20 d^{2}}}{2 a}$
$= \frac{- b \pm 2 \sqrt{5} d}{2 a}$
Therefore, roots of the given quadratic equation are irrational

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If a,b,c are integers and b2=4(ac+5d2),d∈N, then roots of the quadratic equation ax2+bx+c=0

The correct option is C Irrationalb2=4(ac+5d2)D=b2−4ac=4(ac+5d2)−4ac=4ac+20d2−4ac=20d2Now, x=−b±√D2a=−b±√20d22a=−b±2√5d2aTherefore, roots of the given quadratic equation are irrational

If $a, b, c$ are integers and $b^{2} = 4 (a c + 5 d^{2}), d \in N$ , then roots of the quadratic equation $a x^{2} + b x + c = 0$