Question

x=2.414 then find (x - 1÷x)^2

Answer 1

Answer:

$(x-\frac{1}{x})^2=4$

Step-by-step explanation:

Formula used:

$(a+b)(a-b)=a^2-b^2$

Given:

$x=2.414$

$x=1.414+1$

$x=1.414+1$

$x=\sqrt{2}+1$

consider

$\frac{1}{x}=\frac{1}{\sqrt{2}+1}$

To rationalize the denominator, multiply both

numerator and denominator by $\sqrt2-1$

$\frac{1}{x}=\frac{1}{\sqrt{2}+1}*\frac{\sqrt{2}-1}{\sqrt{2}-1}$

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

$\frac{1}{x}=\frac{\sqrt{2}-1}{2-1}$

$\frac{1}{x}=\frac{\sqrt{2}-1}{1}$

$\frac{1}{x}=\sqrt{2}-1$

$x-\frac{1}{x}=(\sqrt{2}+1)-(\sqrt{2}-1)$

$x-\frac{1}{x}=\sqrt{2}+1-\sqrt{2}+1$

$x-\frac{1}{x}=2$

$(x-\frac{1}{x})^2=2^2=4$

Answer 2

We know that the formula of (x - 1/x)² is given by the formula,

     (x - 1/x)² = x² + 1/x² - 2

Now,   x = 2.414

It can be written as, 1.414 + 1

Now, √2 = 1.414,

Thus, x = √2 + 1

Thus, x² = (√2 + 1)

= 2 + 1 + 2√2

= 3 + 2√2

Similarly, 1/x² = 1/(√2 + 1)²

∴ 1/x² = 1/(3 + 2√2)

Thus, (x - 1/x)² = x² + 1/x² - 2

= 3 + 2√2 + 1/(3 + 2√2) - 2

=  [(3 + 2√2)² + 1 - 2(3 + 2√2)]/(3 + 2√2)

= [9 + 8 + 12√2 + 1 - 6 - 4√2]/(3 + 2√2)

= [12 + 8√2]/(3 + 2√2)

Taking 4 common the the numerator,

= 4[3 + 2√2]/(3 + 2√2)

= 4

Hence, the value of the expression is 4.

Hope it helps.