If one of the roots of the equation x2 minus bx c 0 is the square of the other then which of the fol...........

Quantitative Aptitude (CDS) Quadratic Equations

If one of the roots of the equation x² − bx+ c = 0 is the square of the other, then which of the following option is correct?

b³ = 3bc + c² + c

c³ = 3bc+ b² + b

3bc=c³+b²+b

3bc=c³+b³+b²

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Explanation:

According to equation, let one root of equation is $\alpha$ then other roots of equation is $\alpha$ ²
Sum of roots = $\alpha$ + $\alpha$ ²= - (- b/1)
$\alpha$ ( $\alpha$ + 1) = b ………. (i)
Product of root = $\alpha$ x $\alpha$ ² = (c/1)
$\alpha$ ³ = c ⇒ $\alpha$ = c^1/3 ………. (ii)
From (i) and (ii)
c^1\3(c^1\3 +1) = b ……….(iii)
On cubing both sides, we get
c (c^1\3 + 1)³ = b³
⇒ c(c+1+3c^2/3+3c^1/3) = b³