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Nature of Roots

If the roots ...

Question

If the roots of the equation $a x^{2} + b x + c = 0$ are real and distinct. Then

A

Both roots are greater than

- \frac{b}{2 a}

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B

Both roots are less than

- \frac{b}{2 a}

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C

One of the roots exceeds

- \frac{b}{2 a}

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D

None of the above

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Solution

The correct option is B One of the roots exceeds $- \frac{b}{2 a}$
Let $α$ and $β$ be the roots of the equation $a x^{2} + b x + c = 0$
The roots of the given equation are
$α = \frac{- b - \sqrt{b^{2} - 4 a c}}{2 a}$
and $β = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}$
Since, $α, β$ are real and distinct, therefore
$b^{2} - 4 a c > 0$ . Now, if $a > 0$ , then $β > - \frac{b}{2 a}$ and if $a < 0$ , then $α > - \frac{b}{2 a}$ .
Thus, one of the roots exceeds $- \frac{b}{2 a}$ .

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Similar questions

Q. (a) If the roots

α, β

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

be real, then establish between the coefficients under the following conditions:
(i) Roots are equal and opposite.
(ii) Roots are of opposite signs.
(iii) Roots are both

- i v e

(iv) Roots are both

+ i v e

a > 0, b > 0

, then both roots of the equation

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

(i) are real and negative
(ii) have negative real parts.
(iii) none of these.
(c) If the roots of the equation

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

be imaginary. then for all real values of x, the expression

3 b^{2} x^{2} + 6 b c x + 2 c^{2}

is
(a) less than

- 4 a b

(b) greater then

4 a b

4 a b

(d) greater than

- 4 a b

Q. If both roots of

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

are greater than

k

- \frac{b}{2 a} > k .

For the expression f(x) = a $x^{2}$ + bx + c (a > 0), having both real roots, the condition for both real roots to be greater than a real value $x_{0}$ is

a, b, c

a \neq b

, the roots of the equation

(a - b) x^{2} - 2 (a + b) x + (a - b) = 0

Q. Prove that the roots of

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

\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

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