Name: Condition for Common Roots of Quadratic Equation
Uploaded: 2019-07-19
Duration: 36 min 41 s
Description: In this video, we will learn the condition for one common root and the condition for both roots common for two given quadratic equations with the help of exa...

Question

If

x_{1}

is a root of

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

,

x_{2}

is a root of

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

where

0 < x_{1} < x_{2}

, show that the equation

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

has a root

x_{3}

satisfying

0 < x_{1} < x_{3} < x_{2}

.

Accepted Answer

Since $x_{1}$ is root of quadratic equation $a x^{2} + b x + c = 0$

$\Rightarrow a {x_{1}}^{2} + b x_{1} + c = 0$

$\Rightarrow b x_{1} + c = - a {x_{1}}^{2} . . . . . . . . . . . (i)$

Since $x_{2}$ is root of quadratic equation $- a x^{2} + b x + c = 0$ .

$\Rightarrow - a {x_{2}}^{2} + b x_{2} + c = 0$

$\Rightarrow b x_{2} + c = a {x_{2}}^{2} . . . . . . . . . . . (ii)$

Now, for $f (x) = a x^{2} + 2 b x + 2 c$ .

$\Rightarrow f (x_{1}) = a {x_{1}}^{2} + 2 b x_{1} + 2 c$

$\Rightarrow f (x_{1}) = a {x_{1}}^{2} + 2 (b x_{1} + c)$

From equation $(i)$ ,

$\Rightarrow f (x_{1}) = a {x_{1}}^{2} - 2 a {x_{1}}^{2}$

$\Rightarrow f (x_{1}) = - a {x_{1}}^{2}$

Also $f (x_{2}) = a {x_{2}}^{2} + 2 b x_{2} + 2 c$

$\Rightarrow f (x_{2}) = a {x_{2}}^{2} + 2 (b x_{2} + c)$

From equation $(ii)$ ,

$\Rightarrow f (x_{2}) = a {x_{2}}^{2} + 2 a {x_{2}}^{2}$

$\Rightarrow f (x_{2}) = 3 a {x_{2}}^{2}$

Clearly if $a > 0 \Rightarrow f (x_{1}) < 0 & f (x_{2}) > 0$ .

If $a < 0 \Rightarrow f (x_{1}) > 0 & f (x_{2}) < 0$

Since $f (x_{1}) & f (x_{2})$ have opposite sign one root lie between $x_{1} & x_{2}$ .

$\Rightarrow 0 < x_{1} < x_{3} < x_{2}$ .

If $x_{1}$ is a root of $a x^{2} + b x + c = 0$ , $x_{2}$ is a root of $- a x^{2} + b x + c = 0$ where $0 < x_{1} < x_{2}$ , show that the equation $a x^{2} + 2 b x + 2 c = 0$ has a root $x_{3}$ satisfying $0 < x_{1} < x_{3} < x_{2}$ .

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If x1 is a root of ax2+bx+c=0, x2 is a root of -ax2+bx+c=0 where 0<x1<x2, show that the equation ax2+2bx+2c=0 has a root x3 satisfying 0<x1<x3<x2.

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If $x_{1}$ is a root of $a x^{2} + b x + c = 0$ , $x_{2}$ is a root of $- a x^{2} + b x + c = 0$ where $0 < x_{1} < x_{2}$ , show that the equation $a x^{2} + 2 b x + 2 c = 0$ has a root $x_{3}$ satisfying $0 < x_{1} < x_{3} < x_{2}$ .