Question

without determining the value of x, evaluate x² + ¹/x² if x - ¹/x = 5​

Answer 1

We know $x - \frac{1}{x} = 5$ .

Notice that the x the question wants is squared, and also notice that when we multiply x and 1/x, they cancel out. So by squaring both sides of equations and manipulating it:

$(x - \frac{1}{x})^2 = 5^2\\\\(x^2 - 2 *x*(-\frac{1}{x}) + \frac{1}{x^2}) = 25\\\\x^2 + 2 +\frac{1}{x^2} = 25\\\\x^2 +\frac{1}{x^2} = 23$

Answer 2

If  $x-\frac{1}{x}=5$

$x^{2} +\frac{1}{x^{2} } =?$

$x-\frac{1}{x}=5$

$(x-\frac{1}{x})^{2} =5^{2}$

$x^{2} +2.x.(-\frac{1}{x}) +(-\frac{1}{x} )^{2} =25$

$x^{2} -\frac{2x}{x} +\frac{1^{2} }{x^{2} } =25$

$x^{2} -2+\frac{1 }{x^{2} } =25$

$x^{2} +\frac{1 }{x^{2} } =25+2$

$x^{2} +\frac{1 }{x^{2} } =27$

27