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(B) Given the equation: $z\bar{z}^3 + \bar{z}z^3 = 350$ Factor out $z\bar{z}$: $z\bar{z}(\bar{z}^2 + z^2) = 350$ Since $z\bar{z} = |z|^2 = x^2 + y^2$,

Vedclass · Accepted Answer

$5$

Vedclass · Answer

(B) Given the equation: $z\bar{z}^3 + \bar{z}z^3 = 350$ Factor out $z\bar{z}$: $z\bar{z}(\bar{z}^2 + z^2) = 350$ Since $z\bar{z} = |z|^2 = x^2 + y^2$,we have: $|z|^2((x - iy)^2 + (x + iy)^2) = 350$ Expand the squares: $|z|^2(x^2 - y^2 - 2xyi + x^2 - y^2 + 2xyi) = 350$ $|z|^2(2x^2 - 2y^2) = 350$ $2|z|^2(x^2 - y^2) = 350$ $|z|^2(x^2 - y^2) = 175$ Since $|z|^2 = x^2 + y^2$,we have $(x^2 + y^2)(x^2 - y^2) = 175$. $x^4 - y^4 = 175$. Testing integer values for $x$ and $y$: If $x = 4, y = 3$,then $4^4 - 3^4 = 256 - 81 = 175$. Thus,$|z|^2 = x^2 + y^2 = 4^2 + 3^2 = 16 + 9 = 25$. Therefore,$|z| = \sqrt{25} = 5$.

If $z = x + iy$ is a complex number such that $z\bar{z}^3 + \bar{z}z^3 = 350$ and $x, y$ are integers,then $|z| = $

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