Solving of Algebraic Equation: (1019-x)^(0.5) - (2019+x)^(1/3) = 16

Question

The first part of the Left Hand Side is a square root and the second part of it is a cube root. Its all about finding 'x' from this equation....

(1019-x)^(0.5) - (2019+x)^(1/3) = 16

Answer 1

Solving this equation using Newton's iterative method tells us that the answer is 178.

Let's see if it can be solved analytically?

Let's write this as (1019-x)^(1/2) = 16 + (2019+x)^(1/3) Let's square both sides... then

1019-x = 256 + 32(2019+x)^(1/3) + (2019+x)^(2/3) Let z = (2019+x)^(1/3), then x = z³ - 2019

This transforms the equation to

1019 -z³ + 2019 = 256 +32z + z² or z³ + z² + 32z - 2782 = 0

Using an online cubic equation solver, we find the only real z root is 13.

so x = (13)³ - 2019 = 178.