Quadratic? Complete the square? Fractional Exponents Math help needed?

Question

I need help with two math problems

First : x^1/2 + x ^1/4 -6 = 0

Second: x^2 +(x-4)^2 = 10^2

Any help is greatly appreciated.

Ive done these problems a few times and im unsure.

Answer 1

First : x^1/2 + x ^1/4 -6 = 0

Let x^(1 / 4) = u.

Then:

u^2 + u - 6 = 0

You don't need to complete the square for this one.

(u + 3)(u - 2) = 0

u = - 3 or u = 2

x = +/- i sqrt(3) or +/- sqrt(2).

Second: x^2 +(x-4)^2 = 10^2

Expanding the bracket:

x^2 + x^2 - 8x + 16 = 100

2x^2 - 8x - 84 = 0

x^2 - 4x - 42 = 0

x^2 - 4x = 42

Here you do need to complete the square or use the quadratic formula.

x^2 - 4x + 4 = 46

(x - 2)^2 = 46

x - 2 = +/- sqrt(46)

x = 2 +/- sqrt(46).