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Question Number 202353 by MATHEMATICSAM last updated on 25/Dec/23

If 2x = a + (√((4b^3 − a^3 )/(3a))) and 2y = a − (√((4b^3 − a^3 )/(3a))) then show that x^3 + y^3 = b^3 .

If2x=a+4b3−a33aand2y=a−4b3−a33athenshowthatx3+y3=b3.

Answered by esmaeil last updated on 25/Dec/23

x+y=a( i) xy=(a^2 /4)+((a^3 −4b^3 )/(12a))=((a^3 −b^3 )/(3a)) (ii) i ,ii→x^2 +y^2 =a^2 +((2b^3 −2a^3 )/(3a))=((a^3 +2b^3 )/(3a)) x^3 +y^3 = (x+y)(x^2 +y^2 −xy)= a(((a^3 +2b^3 )/(3a))+((b^3 −a^3 )/(3a)))=b^3

x+y=a(i)xy=a24+a3−4b312a=a3−b33a(ii)i,ii→x2+y2=a2+2b3−2a33a=a3+2b33ax3+y3=(x+y)(x2+y2−xy)=a(a3+2b33a+b3−a33a)=b3

Answered by Rasheed.Sindhi last updated on 25/Dec/23

x+y=a & x−y=(√((4b^3 − a^3 )/(3a))) 4xy=(x+y)^2 −(x−y)^2 =a^2 −((4b^3 − a^3 )/(3a))=((3a^3 +a^3 −4b^3 )/(3a)) xy=((a^3 −b^3 )/(3a)) • x^3 +y^3 =(x+y)( (x+y)^2 −3xy ) =(a)( a^2 −3(((a^3 −b^3 )/(3a))) ) =a^3 −(a^3 −b^3 )=b^3 ✓

x+y=a&x−y=4b3−a33a4xy=(x+y)2−(x−y)2=a2−4b3−a33a=3a3+a3−4b33axy=a3−b33a∙x3+y3=(x+y)((x+y)2−3xy)=(a)(a2−3(a3−b33a))=a3−(a3−b3)=b3✓

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