Question

(x+16)^1/2=x-(x+17)^1/2

Answer 1

Given the equation

`√(x+16)=x-√(x+17)`

take the square of both sides then move the square root term to one side to get

`\left(√(x+16)\right)^2=\left(x-√(x+17)\right)^2`

`x+16=x^2-2x√(x+17)+(x+17)`

`x^2-2x√(x+17)+1=0`

`x^2+1=2x√(x+17)`

Now take the square of both sides again to get

`\left(x^2+1\right)^2=\left(2x√(x+17)\right)^2`

`x^4+2x^2+1=4x^2(x+17)`

`x^4+2x^2+1=4x^3+68x^2`

`x^4-4x^3-66x^2+1=0`

Use a calculator to find 4 solutions,

`x\approx-6.365`

`x\approx-0.124`

`x\approx0.123`

`x\approx10.366`

Plugging each of these back into the original equation, you would find that the first 3 solutions are extraneous, so the only solution is

`\boxed{x\approx10.366}`