Question

if x-1/x = 2 , find x^4 + 1/x^4​

Answer 1

Answer:

If x+ 1/x=2,then the value of x^4 -1/x^4 is?

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So the given equation is, x+1/x=2.

Thus, 2x=x+1.

Hence, x=1.

I hope we are clear till here.

The question asked is x^4 - 1/x^4.

Putting the value of x that we obtained from the given equation(i.e. x=1) in the second equation we get,

1^4 - 1/1^4 = 1 -1/1 = 1-1 = 0.

Looks like the above solution is for the equation:

[ (x+1)/x] = 2

No problem….I will write the other solution too.

The above equation can also be perceived as

x + (1/x) = 2

So, in this case we multiply LHS and RHS with X.

We get,

x^2 + 1 = 2x

~ x^2 - 2x + 1 = 0

Applying the Shridhar Acharya Formula,

i.e. x = [{ -b + √D}/2a] or x = [{ -b - √D}/2a]

where

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The problem is very easy: if x+1x=2 , then x2–2x+1=0 and so x=1 .

However, the problem can be made much more interesting by generalizing it: suppose x+1x=a and determine the value of x4−1x4 .

Let’s write y=1/x for simplicity, so we can always substitute xy=1 .

The assumption implies (x+y)2=a2 and so x2+2xy+y2=a2 and so x2+y2=a2−2 .

You want to find the value of x4−y4=(x2+y2)(x+y)(x−y) , so we need to find x−y . However, (x−y)2=x2–2xy+y2=a2–4 . If we assume x≥1 , then x>y and so

x−y=a2–4−−−−√

and we see that we need a≥2 (which also follows from the “arithmetic mean–geometric mean inequality”).

Final answer: if x+1x=a and x≥1 then

x4−1x4=a(a2–2)a2–4−−−−√

If 0<x<1 , then the expression just changes sign.

Step-by-step explanation:

I hope it will be helpful for you.

Answer 2

Given,

$\boxed{ \huge \bf{x - \frac{1}{x} = 2}}$

• To find : x⁴ + 1/x⁴ = ?

$\overline{ \underline{ \Huge \mathbb{ \dag \: \: SOLUTION \: \: \dag}}}$

$\rm{}x - \frac{1}{x} = 2 \\ \small \boxed{ \tt{sqauring \: \: on \: \: both \: \: sides}} \\ \rm {\bigg(x - \frac{1}{x} \bigg) }^{2} = {(2)}^{2}$

• Identity Required : (a-b)² = a² -2ab+ b²

$\to \rm {x}^{2} + \frac{1}{ {x}^{2} } - 2\bigg( \cancel{x}\bigg)\bigg( \frac{1}{\cancel{x}} \bigg) = 4 \\ \\ \to \rm{x}^{2} + \frac{1}{ {x}^{2} } - 2 = 4 \\ \small \boxed{ \tt{transposing \: \: - 2 \: \: to \: \: RHS}} \\ \to \rm {x}^{2} + \frac{1}{ {x}^{2} } = 4 + 2 \\ \\ \to \rm {x}^{2} + \frac{1}{ {x}^{2} } = \orange6$

Now,

• Squaring both sides of obtained expression, we get,

$\to \rm { \bigg( {x}^{2} + \frac{1}{ {x}^{2} } \bigg)}^{2} = {(6)}^{2}$

• Identity Required : (a+b)² = a² +2ab+ b²

$\to \rm {( {x}^{2} )}^{2} + \frac{1}{ {( {x}^{2} )}^{2} } + 2 \bigg( \cancel{{x}^{2} } \bigg) \bigg( \frac{1}{ \cancel{{x}^{2} } } \bigg) = 36 \\ \\ \to \rm {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 36 \\ \small \boxed{ \tt{transposing \: \: 2 \: \: to \: \:RHS }} \\ \to \rm {x}^{4} + \frac{1}{ {x}^{4} } = 36 - 2 \\ \\ \to \rm {x}^{4} + \frac{1}{ {x}^{4} } = \sf \red{34}$

Thus,

• We obtained the value of x⁴ + 1/x⁴ as 34.