Cubic and Biquadratic equation

Algebra Level 4

Find the relation between q q and r r in order that the equation x 3 + q x + r = 0 x^{3}+qx+r=0 may be put into form x 4 = ( x 2 + a x + b ) 2 x^{4}=(x^{2}+ax+b)^{2}

q 3 + 8 r 2 = 0 q^{3}+8r^{2}=0 8 q 2 = r 3 8q^{2}=r^{3} 2 q 2 = r 3 2q^{2}=r^{3} q 2 + r 3 = 0 q^{2}+r^{3}=0

Aditya Lalbondre by Aditya Lalbondre , 22, India

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1 solution

Tom Engelsman
Oct 24, 2020

If we expand the second equation out, one obtains:

x 4 = x 4 + 2 x 2 ( a x + b ) + ( a x + b ) 2 ; x^4 = x^4 + 2x^2(ax+b) + (ax+b)^2;

or 0 = 2 a x 3 + ( a 2 + 2 b ) x 2 + 2 a b x + b 2 0 = 2ax^3 +(a^2 + 2b)x^2 + 2abx + b^2 .

In order to put this cubic into the form x 3 + q x + r = 0 x^3 + qx + r = 0 we require:

2 a = 1 ⇒ a = 1 2 ; 2a = 1 \Rightarrow a = \frac{1}{2};

a 2 + 2 b = 0 ⇒ b = − 1 8 a^2 + 2b = 0 \Rightarrow b = -\frac{1}{8}

which gives us x 3 − 1 8 x + 1 64 = 0 ⇒ q = − 1 8 , r = 1 64 x^3 - \frac{1}{8}x + \frac{1}{64} = 0 \Rightarrow q = -\frac{1}{8}, r = \frac{1}{64} and satisfies q 3 + 8 r 2 = 0 . \boxed{q^3 + 8r^2 = 0}.

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