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

a, b, c

represents sides of a

∆

then prove that equation

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

has imaginary roots.

Accepted Answer

We have to prove that roots of quadratic equation $x^{2} - (a^{2} + b^{2} + c^{2}) x + a^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2} = 0$ are imaginary.

$\Rightarrow D = {(- (a^{2} + b^{2} + c^{2}))}^{2} - 4 (1) (a^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2})$

$\Rightarrow D = {(a^{2} + b^{2} + c^{2})}^{2} - 4 (a^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2})$

$\Rightarrow D = a^{4} + b^{4} + c^{4} + 2 a^{2} b^{2} + 2 b^{2} c^{2} + 2 c^{2} a^{2} - 4 a^{2} b^{2} - 4 b^{2} c^{2} - 4 c^{2} a^{2}$

$\Rightarrow D = a^{4} + b^{4} + c^{4} - 2 a^{2} b^{2} - 2 b^{2} c^{2} - 2 c^{2} a^{2}$

Add and subtract $2 c^{2} a^{2}$

$\Rightarrow D = a^{4} + b^{4} + c^{4} - 2 a^{2} b^{2} - 2 b^{2} c^{2} - 2 c^{2} a^{2} + 2 c^{2} a^{2} - 2 c^{2} a^{2}$

$\Rightarrow D = a^{4} + b^{4} + c^{4} - 2 a^{2} b^{2} - 2 b^{2} c^{2} + 2 c^{2} a^{2} - 4 c^{2} a^{2}$

$\Rightarrow D = {(a^{2} + c^{2} - b^{2})}^{2} - 4 c^{2} a^{2}$

$\Rightarrow D = (a^{2} + c^{2} - b^{2} - 2 a c) (a^{2} + c^{2} - b^{2} + 2 a c)$

$\Rightarrow D = ({(a - c)}^{2} - b^{2}) ({(a + c)}^{2} - b^{2})$

Since $a, b, c$ are sides of triangle $\Rightarrow a + c > b & |a - c| < b$

$\Rightarrow {(a + c)}^{2} > b^{2} & {(a - c)}^{2} < b^{2}$

$\Rightarrow D < 0 \Rightarrow$ Roots are imaginary.

If $a, b, c$ represents sides of a $∆$ then prove that equation $x^{2} - (a^{2} + b^{2} + c^{2}) x + a^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2} = 0$ has imaginary roots.

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If a, b, c represents sides of a ∆ then prove that equation x2-a2+b2+c2x+a2b2+b2c2+c2a2=0 has imaginary roots.

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If $a, b, c$ represents sides of a $∆$ then prove that equation $x^{2} - (a^{2} + b^{2} + c^{2}) x + a^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2} = 0$ has imaginary roots.