If the roots of the equation ax2-2bx-c-0 and bx2-2-acx-b-0 are simultaneously real then prove that b2-ac-

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

Location of Roots

If the roots ...

Question

If the roots of the equation $a x^{2} + 2 b x + c = 0$ and $b x^{2} - 2 \sqrt{a c} x + b = 0$ are simultaneously real then prove that $b^{2} = a c$ .

Open in App

Solution

Given, the roots of both the equations are real.

First equation:

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

Its discriminant, $D \geq 0$

⇒ $(2 b)^{2} - 4 (a c) \geq 0$

⇒ $4 b^{2} - 4 a c \geq 0$

⇒ $4 b^{2} \geq 4 a c$

⇒ $b^{2} \geq a c$ ... (1)

Second equation:

$b x^{2} - 2 \sqrt{a c} x + b = 0$

Its discriminant, $D \geq 0$

⇒ $(2 \sqrt{a c})^{2} - 4 (b^{2}) \geq 0$

⇒ $4 a c - 4 b^{2} \geq 0$

⇒ $4 a c \geq 4 b^{2}$

⇒ $a c \geq b^{2}$ ... (2)

The results of equation (1) and (2) are simultaneously possible in only one case when $b^{2} = a c .$

Suggest Corrections

103

Similar questions

Q. If the roots of the equations

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

and

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

are simultaneously real then prove that

b^{2} = a c

Q. If roots of the equations

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

and

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

are real, then

Q. The roots of the equation

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

are real if those of

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

are imaginary.

If $α, β$ are the roots of the equation a $x^{2}$ +2bx +c = 0 and $α + δ$ , $β + δ$ are the roots of A $x^{2}$ + 2Bx + C= 0 , then $\frac{b^{2} - a c}{B^{2} - A C}$ =

If α, β are the roots of a $x^{2}$ +2bx+c=0 and a+δ, β+δ are the roots of A $x^{2}$ + 2Bx + C=0 then $\frac{b^{2} - a c}{B^{2} - A c}$ is

Join BYJU'S Learning Program

If the roots of the equation ax2+2bx+c=0 and bx2−2√acx+b=0 are simultaneously real then prove that b2=ac.

If the roots of the equation $a x^{2} + 2 b x + c = 0$ and $b x^{2} - 2 \sqrt{a c} x + b = 0$ are simultaneously real then prove that $b^{2} = a c$ .