Question

if x sq. + 1/x sq. = 23, find x + 1/x

Answer 1

Answer:

If x^2 + 1/(x^2) = 23, find x?

Let x^2 + 1/(x^2) = 23 be >>> Equation (1)

It would be helpful if we could find an expression which gave us all or part of Equation (1).

Let us consider

( x + 1/x )^2 >>> Equation (2)

Expanding Equation (2) we get:

x^2 + 2 + 1/( x )^2 >>>>> Equation (3)

Therefore re-arranging Equation(3)

[( x^2 + 1/(x)^2)] + 2 is the same as 23 + 2 = 25.

That is [ x^2 + 1/( x )^2 ] = 25 >>>>>> Equation (4)

Now take the square root of both sides of Equation (4).

[ x + 1/x ] = sqrt ( 25 ) or

(x + 1/x) = (+\-) 5 >> Equation (5)

Let us reformat Equation (5) by multiplying every term by x.

[ x^2 + 1 ] = (+\-)(5x) >>> Equation (6)

Re-arranging Equation (6)

x^2 (+\-)5x + 1= 0 >>>> Equation (7)

Case (1) +5x

Roots = [ -5 (-\+) sqrt (25 - 4) ]/2 or

Roots are: -4.7913 and -0.20871

Case (2) -5x

Roots = [ +5 (-\+)sqrt (25 - 4) ]/2 or

Roots are +4.7913 and +0.20871

It would have been nice to find the roots were nice clean integers. But apparently not!

So to check, I graphed the original equation as below and confirmed that the analysis was correct.