For the quadratic equation ax2-bx-c-0- a- b- c- if it-s discriminant D is square of a non zero rational number- then it-s roots will be

Given the quadratic equation ax^2+bx+c=0; a, b, c∈ℚ Now, the roots of the equation are given by: x= b±√(b^2 4ac)2a Or x= b±√(D)2a Now, D is the square of a rati ...

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Standard XII

Mathematics

Methods of Solving a Quadratic Equation

For the quadr...

Question

For the quadratic equation $a x^{2} + b x + c = 0; a, b, c \in Q$ if it's discriminant $D$ is square of a non zero rational number, then it's roots will be

rational & unequal

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irrational & unequal

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imaginary & unequal

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Solution

The correct option is A rational & unequal
Given the quadratic equation $a x^{2} + b x + c = 0; a, b, c \in Q$
Now, the roots of the equation are given by:
$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} Or x = \frac{- b \pm \sqrt{D}}{2 a}$
Now, $D$ is the square of a rational number
$\Rightarrow \sqrt{D} = \frac{p}{q} \neq 0, q \neq 0$
Substituting in the equation, we get:
$x = \frac{- b \pm \frac{p}{q}}{2 a} \Rightarrow x = \frac{- b + \frac{p}{q}}{2 a} or x = \frac{- b - \frac{p}{q}}{2 a} \Rightarrow x = \frac{- b q + p}{2 a q} or x = \frac{- b q - p}{2 a q}$
Which are rational numbers.
Hence, the roots for such quadratic equation will be rational and distinct numbers.

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For the quadratic equation ax2+bx+c=0; a, b, c∈Q if it's discriminant D is square of a non zero rational number, then it's roots will be

For the quadratic equation $a x^{2} + b x + c = 0; a, b, c \in Q$ if it's discriminant $D$ is square of a non zero rational number, then it's roots will be