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-

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

If 2 x = a + \sqrt{\frac{4 b^{3} - a^{3}}{3 a}} and

2 y = a - \sqrt{\frac{4 b^{3} - a^{3}}{3 a}} then show that

x^{3} + y^{3} = b^{3} .

Answered by esmaeil last updated on 25/Dec/23

x + y = a (i)

x y = \frac{a^{2}}{4} + \frac{a^{3} - 4 b^{3}}{12 a} = \frac{a^{3} - b^{3}}{3 a} (i i)

i, i i \to x^{2} + y^{2} = a^{2} + \frac{2 b^{3} - 2 a^{3}}{3 a} = \frac{a^{3} + 2 b^{3}}{3 a}

x^{3} + y^{3} =

(x + y) (x^{2} + y^{2} - x y) =

a (\frac{a^{3} + 2 b^{3}}{3 a} + \frac{b^{3} - a^{3}}{3 a}) = b^{3}

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

x + y = a & x - y = \sqrt{\frac{4 b^{3} - a^{3}}{3 a}}

4 x y = {(x + y)}^{2} - {(x - y)}^{2}

= a^{2} - \frac{4 b^{3} - a^{3}}{3 a} = \frac{3 a^{3} + a^{3} - 4 b^{3}}{3 a}

x y = \frac{a^{3} - b^{3}}{3 a}

∙ x^{3} + y^{3} = (x + y) ({(x + y)}^{2} - 3 x y)

= (a) (a^{2} - 3 (\frac{a^{3} - b^{3}}{3 a}))

= a^{3} - (a^{3} - b^{3}) = b^{3} ✓