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Nature of Roots

If a, b, c ...

Question

If $a, b, c$ are real and $b^{2} - 4 a c$ is perfect square then the roots of the equation $a x^{2} + b x + c = 0$ , will be:

Rational & distinct

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Real & equal

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Irrational & distanct

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Imaginary & distinct

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Solution

The correct option is B Rational & distinct
Given that the discriminant $= b^{2} - 4 a c$ is a perfect square

Let $b^{2} - 4 a c = k^{2}$

$\Rightarrow$ Roots $= \frac{- b \pm \sqrt{k^{2}}}{2 a}$

$\Rightarrow$ roots $= \frac{- b \pm k}{2 a}$

As given $a, b, c$ are real
$\Rightarrow$ Roots are rational and distinct.

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Similar questions

Q. For the quadratic equation

a x^{2} + b x + c = 0, a, b

and

c

are rational numbers and

D = b^{2} - 4 a c

is a non-zero perfect square. The roots of the equation are

Q. The roots of

a x^{2} + b x + c = 0, a \neq 0

are real and unequal, if

b^{2} - 4 a c

is ___________.

Q. 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

The roots of a quadratic equation $a x^{2} + b x + c = 0$ are given by $\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , provided $b^{2} - 4 a c \geq 0$ .

Q. Assertion :If

a, b, c ϵ Z

and

a x^{2} + b x + c = 0

has an irrational root, then

| f (λ) | \geq 1 / q^{2}

, where

λ ϵ (λ = \frac{p}{q}; p, q ϵ Z)

and

f (x) = a x^{2} + b x + c

. Reason: If

a, b, c ϵ Q

and

b^{2} - 4 a c

is positive but not a perfect square, then roots of equation

a x^{2} + b x + c = 0

are irrational and always occur in conjugate pair like

2 + \sqrt{3}

and

2 - \sqrt{3}

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If a,b,c are real and b2−4ac is perfect square then the roots of the equation ax2+bx+c=0, will be:

The correct option is B Rational & distinctGiven that the discriminant =b2−4ac is a perfect square Let b2−4ac=k2 ⇒ Roots =−b±√k22a⇒ roots =−b±k2aAs given a,b,c are real ⇒ Roots are rational and distinct.

If $a, b, c$ are real and $b^{2} - 4 a c$ is perfect square then the roots of the equation $a x^{2} + b x + c = 0$ , will be: