In the equation ax^2 +bx+c=0 one root is square of orther.without solving the equation.prove that c(a−b)^3 =a(c−b)^3


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Question Number 48193 by peter frank last updated on 20/Nov/18

$I n t h e e q u a t i o n {a x}^{2} + b x + c = 0$ $o n e r o o t i s s q u a r e o f$ $o r t h e r . w i t h o u t s o l v i n g$ $t h e e q u a t i o n . p r o v e t h a t$ $c {(a - b)}^{3} = a {(c - b)}^{3}$
	Answered by MJS last updated on 20/Nov/18

	$a (x - α) (x - α^{2}) = {a x}^{2} - a α (α + 1) x + a α^{3}$ $a α^{3} {(a + a α (α + 1))}^{3} = a^{4} α^{3} {(α^{2} + α + 1)}^{3}$ $a {(a α^{3} + a α (α + 1))}^{3} = a^{4} α^{3} {(α^{2} + α + 1)}^{3}$
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