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

Question

asked Dec 24, 2021 106k views

1 Answer

Cejast · Answer 1 · 2021-12-31T02:52:06+0000

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

$\boxed{x\approx10.366}$