Question

 If x + (1/x) = 5, find the value of [x - (1/x)]^2.​

Answer 1

Step-by-step explanation:

Given :-

x+(1/x) = 5

To find :-

Find the value of [x - (1/x)]²?

Solution :-

Method-1:-

Given that

x+(1/x) = 5

On squaring both sides then

=> [x+(1/x)]² = 5²

=> x²+(1/x)²+2(x)(1/x) = 25

Since (a+b)² = a²+2ab+b²

=> x²+(1/x²)+2(x/x) = 25

=> x²+(1/x²)+2(1) = 25

=> x²+(1/x²)+2 = 25

=> x²+(1/x²) = 25-2

=> x²+(1/x²) = 23 --------(1)

Now,

The value of [x - (1/x)]²

=> x²+(1/x)²-2(x)(1/x)

Since (a-b)² = a²-2ab+b²

=> x²+(1/x²) -2(x/x)

=> x²+(1/x²) -2(1)

=> x²+(1/x²)-2

=> 23-2

=> 21

Therefore, [x - (1/x)]² = 21

Method-2:-

Given that

x+(1/x) = 5

We know that

(a-b)² = (a+b)²-4ab

On taking a = x and b = 1/x

Now,

[x-(1/x)]² = [x+(1/x)]²-4(x)(1/x)

=> [x-(1/x)]² = [x+(1/x)]²-4(x/x)

=> [x-(1/x)]² = [x+(1/x)]²-4(1)

=> [x-(1/x)]² = [x+(1/x)]²-4

=> [x-(1/x)]² = (5)²-4

=> [x-(1/x)]² = 25-4

=> [x-(1/x)]² = 21

Therefore, [x - (1/x)]² = 21

Answer:-

The value of [x - (1/x)]² = 21

Used formulae:-

→ (a+b)² = a²+2ab+b²

→ (a-b)² = a²-2ab+b²

→ (a-b)² = (a+b)²-4ab

Answer 2

Answer:

21

Step-by-step explanation:

Given :

$\mathrm{x + \dfrac{1}{x} = 5}$

To find :

$\mathrm{\bigg(x - \dfrac{1}{x}\bigg)^2}$

Solution :

Given condition is,

$\longmapsto \mathrm{x + \dfrac{1}{x} = 5}$

Squaring on both sides

$\implies \mathrm{\bigg(x + \dfrac{1}{x}\bigg)^2 = 5^2}$

We know that,

$\boxed{\mathrm{(a + b)^2 = a^2 + 2ab + b^2}}$

$\implies \mathrm{\bigg(x^2 +2\not x. \dfrac{1}{\not x} + \dfrac{1}{x^2}\bigg) = 25}$

$\implies \mathrm{\bigg(x^2 +2 + \dfrac{1}{x^2}\bigg) = 25}$

$\implies \mathrm{x^2 + \dfrac{1}{x^2} = 25 - 2}$

$\implies \mathrm{x^2 + \dfrac{1}{x^2} = 23}$ ___ [1]

Now, given question is,

$\longmapsto \mathrm{\bigg(x - \dfrac{1}{x}\bigg)^2}$

We know that,

$\boxed{\mathrm{(a - b)^2 = a^2 - 2ab + b^2}}$

Here, a = x ; b = 1/x, so,

$\implies \mathrm{\bigg(x^2 -2\not x. \dfrac{1}{\not x} + \dfrac{1}{x^2}\bigg)}$

$\implies \mathrm{x^2 -2+ \dfrac{1}{x^2}}$

$\implies \mathrm{x^2+ \dfrac{1}{x^2} - 2}$

From [1]

$\implies \mathrm{23 - 2}$

$\implies 21$

$\therefore \underline{\boxed{\mathbf{\bigg(x - \dfrac{1}{x}\bigg)^2 = 21}}}$

Hope it helps!!