SOLUTION: If x = ( 7 plus 4 sqrt 3 ) , then the value of x squared plus 1 divided by x squared is -----?

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: If x = ( 7 plus 4 sqrt 3 ) , then the value of x squared plus 1 divided by x squared is -----?      Log On


   

Question 935085: If x = ( 7 plus 4 sqrt 3 ) , then the value of x squared plus 1 divided by x squared is -----?
Found 2 solutions by MathLover1, srinivas.g:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
x+=+%28+7+%2B+4sqrt+%283+%29%29 , then the value of %28x%5E2%2B1%29%2Fx%5E2 is:

%28%28+7+%2B+4sqrt+%283+%29%29%5E2%2B1%29%2F%28+7+%2B+4sqrt+%283+%29%29%5E2

%28%28+7+%2B+4%2A1.73+%29%5E2%2B1%29%2F%28+7+%2B+4%2A1.73%29%5E2

%28%2813.92%29%5E2%2B1%29%2F%28+13.92%29%5E2

%28193.7664%2B1%29%2F%28+193.7664%29

194.7664%2F193.7664

1.005160853481305 which is approximately 1


Answer by srinivas.g(540) About Me  (Show Source):
You can put this solution on YOUR website!
x+=+7%2B4%2Asqrt%283%29

Formula: +%28a%2Bb%29%5E2=a%5E2%2Bb%5E2%2B2ab
here a = x , b =1/x
so +%28x%2B1%2Fx%29%5E2=+x%5E2%2B%281%2Fx%29%5E2%2B2%2Ax%2A1%2Fx
+%28x%2B1%2Fx%29%5E2=+x%5E2%2B1%2Fx%5E2%2B2
move 2 to the right
+%28x%2B1%2Fx%29%5E2+-2+=+x%5E2%2B1%2Fx%5E2....eq(1)
but +x=+7%2B4%2Asqrt%283%29
1%2Fx++=+1%2F%287%2B4%2Asqrt%283%29%29
we have to rationalize the above term
multiply and divide with %287-4%2Asqrt%283%29%29

+1%2Fx+=+%287-4%2Asqrt%283%29%29%2F%28%287%2B4%2Asqrt%283%29%29%2A%287-4%2Asqrt%283%29%29%29
formula: %28p%2Bq%29%2A%28p-q%29=+p%5E2-q%5E2
+1%2Fx+=+%287-4%2Asqrt%283%29%29%2F%287%5E2-%284%2Asqrt%283%29%29%5E2%29
1%2Fx+=+%287-4%2Asqrt%283%29%29%2F%2849-16%2A3%29
1%2Fx+=%287-4%2Asqrt%283%29%29%2F%2849-48%29
1%2Fx+=%287-4%2Asqrt%283%29%29
+x%2B1%2Fx++=++7%2B4%2Asqrt%283%29%2B7-4%2Asqrt%283%29
+x%2B1%2Fx+=+14
put x+1/x = 14 in eq (1)
+14%5E2-2+=x%5E2%2B1%2Fx%5E2
196-2=+x%5E2%2B1%2Fx%5E2
194+=x%5E2%2B1%2Fx%5E2
so +x%5E2%2B1%2Fx%5E2+=194