CameraIcon

SearchIcon

MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Relationship between Zeroes and Coefficients of a Polynomial

If α, β are...

Question

If $α, β$ are real and distinct roots of $a x^{2} + b x - c = 0$ and $p, q$ are real and distinct roots of $a x^{2} + b x - | c | = 0$ , where $(a > 0)$ , then

A

α, β \in (p, q)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

B

α, β \in [p, q]

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

C

p, q \in (α, β)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

D

None of these

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $α, β \in [p, q]$
$α, β$ are real and distinct roots of $a x^{2} + b x - c = 0$ .
Assume $α < β$
Therefore,
$α, β = \frac{- b \pm \sqrt{b^{2} + 4 a c}}{2 a}$
p, q be real and distinct roots of $a x^{2} + b x - | c | = 0$ .
Assume $p < q$
Therefore,
$p, q = \frac{- b \pm \sqrt{b^{2} + 4 a | c |}}{2 a}$
Now, $c \leq | c |$
$\Rightarrow 4 a c \leq 4 a | c |$
$\Rightarrow \sqrt{b^{2} + 4 a c} \leq \sqrt{b^{2} + 4 a | c |}$
$\Rightarrow - b + \sqrt{b^{2} + 4 a c} \leq - b + \sqrt{b^{2} + 4 a | c |}$
$\Rightarrow \frac{- b + \sqrt{b^{2} + 4 a c}}{2 a} \leq \frac{- b + \sqrt{b^{2} + 4 a | c |}}{2 a}$
$\Rightarrow β \leq q$
Also, $- \sqrt{b^{2} + 4 a c} \geq - \sqrt{b^{2} + 4 a | c |}$
$\Rightarrow - b - \sqrt{b^{2} + 4 a c} \geq - b - \sqrt{b^{2} + 4 a | c |}$
$\Rightarrow \frac{- b - \sqrt{b^{2} + 4 a c}}{2 a} \geq \frac{- b - \sqrt{b^{2} + 4 a | c |}}{2 a}$
$\Rightarrow α \geq p$
Hence, $α, β \in [p, q]$ .

Suggest Corrections

thumbs-up

0

Similar questions

α

β

are the roots of the equation

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

\frac{1 - α}{α}

\frac{1 - β}{β}

are the roots of

p x^{2} + q x + r = 0

\frac{q}{p} = k + \frac{b}{c}

, then the value of

k

α, β

are roots of the equation

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

a, b, c

are distinct real values, then

(1 + α + α^{2}) (1 + β + β^{2})

α

β

are the roots of

x^{2} - p x + q = 0

α, β, γ

are the roots of

x^{3} - a x^{2} + b x - c = 0

(α + \sqrt{β})

(α - \sqrt{β})

are the roots of the equation

x^{2} + p x + q = 0

α, β, p

q

are real, then the roots of the equation

(p^{2} - 4 q) (p^{2} x^{2} + 4 p x) - 16 q = 0

Q. Let p and q be real numbers such that p

\neq 0, p^{3} \neq q

p^{3} \neq - q

α

β

are nonzero complex numbers satisfying

α + β = - p

α^{3} + β^{3} = q

, then a quadratic equation having

\frac{α}{β}

\frac{β}{α}

as its roots is -

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Relationship between Zeroes and Coefficients of a Polynomial

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Relationship between Zeroes and Coefficients of a Polynomial

Standard X Mathematics

Join BYJU'S Learning Program

CrossIcon