Find the two real solutions of x^6 - 198x^3 + 1 = 0?

Question

Please advise, thanks.

Hi, would it be possible to work this out even without a calculator?

P.S. thanks for your help :)

Answer 1

x^6 - 198x^3 + 1 = 0?
let w = x^3

x^6 - 198x^3 + 1 = w^2 - 198w + 1

w^2 - 198w + 1= 0
w = [198 +/- sqrt((198)^2 - 4)] / 2
w = [198 +/- sqrt(39200)] / 2
w = [198 +/- 197.99] / 2
w = 0.0101 /2, w = 395.99 /2
w = 0.005, w = 197.995

x^3 = 0.005, x^3 = 197.995
x = (0.005)^(1/3) , x = (197.995)^(1/3)

Answer 2

x^6 - 198x^3 +1 = 0

1. Let x^3 = A

2. Rewrite the equation:
A^2 - 198A +1 = 0

3. Solve above equation:
A^2 -198A + 1 = 0
A^2 - 2*99A + 99^2 -99^2 +1 = 0
(A-99)^2 - 9800 = 0
(A - 99 + 9800^0.5)*(A - 99 - 9800^ 0.5) = 0

So, A = (99 - 9800^0.5) or A = (99 + 9800^0.5)
Notice: 99-9800^0.5 >0

4. Solution:
x = A^(1/3)

So x = (99 -9800^0.5)^(1/3) or x = (99+9800^0.5)^(1/3)

Answer 3

You can use the quadratic formula to solve for x^3. You get:

x^3 = 99 ± 70 sqrt(2)

This gives us:
x = 3 ± 2 sqrt(2)

None of this "plug in sqrt(2.184782723) into your calculator" stuff is necessary.

See:
http://en.wikipedia.org/wiki/Quadratic_formula#Quadratic_formula

Answer 4

x^6 - 198x^3 + 1 = 0
Let u = x^3
u^2 - 198u + 1 = 0
u = [198 +/- sqrt(198^2-4)]/2
u =99 +/- .5(197.98980
u = 99 +/- 98.9949
u = 99-98.995=.0051
u = 99+98.995= 197.9949
x= cube root 197.9949 = 5.828
x = cubroot .0051 = 0.1721

Answer 5

First of all, let n = x^3. Now you can rewrite your equation like this:

n^2 - 198n + 1 = 0

Now, you can use the quadradic formula:

n = [-b +- sqrt(b^2 - 4ac)]/2a
n = [198 +- sqrt(39204 - 4)]/2
n = [198 +- sqrt(39200)]/2
n = (198 +- 197.989899)/2
n = (395.989899)/2 or n = (0.010101)/2
n = 197.994949 or n = 0.005050

So, since n = x^3, x = cuberoot of n, so
x = cuberoot(197.994949) or x = cuberoot(0.005050)
x = 5.828426 or x= 0.171566

Answer 6

Substitute y = x^3, then your eq is:
y^2 - 198*y + 1 = 0
with solutions:
y1 = 197.9949494, and
y2 = 0.005050634
Then:
x1 = y1^(1/3) = 5.828427125, and
x2 = y2^(1/3) = 0.171572875.

Answer 7

give up

Answer 8

.17157288 and 5.8284271