SOLUTION: Find the real solutions of the given equation. Use a calculator to express solutions rounded to two decimal places. {{{x^4 + sqrt(2)x^2 - 2 = 0}}}

Algebra ->  Equations -> SOLUTION: Find the real solutions of the given equation. Use a calculator to express solutions rounded to two decimal places. {{{x^4 + sqrt(2)x^2 - 2 = 0}}}      Log On


   

Question 1208849: Find the real solutions of the given equation. Use a calculator to express solutions rounded to two decimal places.

x%5E4+%2B+sqrt%282%29x%5E2+-+2+=+0
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Make the "change of variable" y=x%5E2 and turn the equation into the quadratic equation
y%5E2+%2B+sqrt%282%29y+-+2+=+0
Then you could use the quadratic formula that says that the solutions to
ay%5E2%2Bby%2Bc=0 are given by y+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
For the equation above a=1, b=sqrt%282%29, and c=-2
You get (rounded), and
y+=+%28-sqrt%282%29+-+sqrt%282%2B8%29%29%2F2=%28-sqrt%282%29+-+sqrt%2810%29%29%2F2=-2.288246(rounded)
At this point, if as I suspect you were not taught about complex and imaginary numbers, you can only list real solutions.
Then, x%5E2=-2.288246 has no real solutions, and your solutions would be
x=sqrt%280.874032%29=0.93 and x=-sqrt%280.874032%29=-0.93
NOTE:
If you are not fond of the quadratic formula, "completing the square" would be an option,
if you were taught to transform something like y%5E2+%2B+sqrt%282%29y+-+2+=+0
into y%5E2+%2B+sqrt%282%29y+=+2+, add %28sqrt%282%29%2F2%29%5E2 to both sides
to get y%5E2+%2B+sqrt%282%29y+%2B%28sqrt%282%29%2F2%29%5E2=+2+%2B%28sqrt%282%29%2F2%29%5E2,
rewrite it as %28y%2Bsqrt%282%29%2F2%29%5E2=5%2F2 and %28y%2Bsqrt%282%29%2F2%29%5E2=10%2F4 for the solutions
y+%2Bsqrt%282%29%2F2+=+%22+%22+%2B-+sqrt%2810%29%2F2 and y+=%28-sqrt%282%29+%2B-+sqrt%2810%29%29%2F2
They really quote that formula as x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ for ax%5E2%2Bbx%2Bc=0,
but the name/letter used for the variable makes no difference.