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Nature of Roots

Let a, b be n...

Question

Let a, b be non-zero real numbers. Which of the following statements about the quadratic equation $a x^{2} + (a + b) x + b = 0$ is necessarily true?
(I) It has at least one negative root.
(II) It has at least one positive root.
(III) Both its roots are real.

(I) and (II) only

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(I) and (III) only

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(II) and (III) only

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All of them

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Solution

The correct option is B (I) and (III) only
Roots of the given equation are,

$= \frac{- (a + b) \pm \sqrt{(a + b)^{2} - 4 a b}}{2 a}$

$= \frac{- (a + b) \pm (a - b)}{2 a}$

We get the roots as,
$- \frac{b}{a}, - 1$
So, both the roots are real and atleast one negative root.
Hence, B is correct.

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Let a, b be non-zero real numbers. Which of the following statements about the quadratic equation ax2+(a+b)x+b=0 is necessarily true?(I) It has at least one negative root.(II) It has at least one positive root.(III) Both its roots are real.

The correct option is B (I) and (III) onlyRoots of the given equation are,=−(a+b)±√(a+b)2−4ab2a=−(a+b)±(a−b)2a We get the roots as,−ba,−1So, both the roots are real and atleast one negative root.Hence, B is correct.

Let a, b be non-zero real numbers. Which of the following statements about the quadratic equation $a x^{2} + (a + b) x + b = 0$ is necessarily true?
(I) It has at least one negative root.
(II) It has at least one positive root.
(III) Both its roots are real.

The correct option is B (I) and (III) only
Roots of the given equation are,

$= \frac{- (a + b) \pm \sqrt{(a + b)^{2} - 4 a b}}{2 a}$

$= \frac{- (a + b) \pm (a - b)}{2 a}$

We get the roots as,
$- \frac{b}{a}, - 1$
So, both the roots are real and atleast one negative root.
Hence, B is correct.