Name: Problem Based on Formation of Quadratic Equation
Uploaded: 2019-07-19
Duration: 5 min 41 s
Description: In this video, we will learn to solve a problem from the chapter Quadratic Equations, where we need to form an equation whose roots are given. Let's watch an...

Question

If the roots of the equation

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

are equal, prove that either

a = 0

or

a^{3} + b^{3} + c^{3} = 3 a b c

.

Accepted Answer

We have,

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

On comparing it with, $A x^{2} + B x + C = 0$ .
Here, $A = (c^{2} - a b), B = - 2 (a^{2} - b c) and C = (b^{2} - a c)$ .
The discriminant of the given equation is given by

$D = {[- 2 (a^{2} - b c)]}^{2} - 4 (c^{2} - a b) (b^{2} - a c)$

$= 4 {(a^{2} - b c)}^{2} - 4 (c^{2} b^{2} - a c^{3} - a b^{3} + a^{2} b c)$

$= 4 (a^{4} + b^{2} c^{2} - 2 a^{2} b c) - 4 (c^{2} b^{2} - a c^{3} - a b^{3} + a^{2} b c)$

$= 4 [a^{4} + b^{2} c^{2} - 2 a^{2} b c - c^{2} b^{2} + a c^{3} + a b^{3} - a^{2} b c]$

$= 4 [a^{4} + a c^{3} + a b^{3} - 3 a^{2} b c]$

$= 4 a (a^{3} + c^{3} + b^{3} - 3 a b c)$

The given equation will have equal roots, if $D = 0$ .

$\Rightarrow 4 a (a^{3} + b^{3} + c^{3} - 3 a b c) = 0$

$\Rightarrow 4 a = 0$ or, $a^{3} + b^{3} + c^{3} - 3 a b c = 0$

$\Rightarrow a = 0$ or, $a^{3} + b^{3} + c^{3} = 3 a b c$

Hence, proved.

If the roots of the equation $(c^{2} - a b) x^{2} - 2 (a^{2} - b c) x + b^{2} - a c = 0$ are equal, prove that either $a = 0$ or $a^{3} + b^{3} + c^{3} = 3 a b c$ .

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If the roots of the equation c2-abx2-2a2-bcx+b2-ac=0 are equal, prove that either a=0 or a3+b3+c3=3abc.

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If the roots of the equation $(c^{2} - a b) x^{2} - 2 (a^{2} - b c) x + b^{2} - a c = 0$ are equal, prove that either $a = 0$ or $a^{3} + b^{3} + c^{3} = 3 a b c$ .