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X ^ 2 + 1/(x ^ 2) = 3 find the value of x+1/x​

User Kajo
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2 votes

Answer:

x + 1/x = ± √((3 ± √5) / 2) + 1 / (√((3 ± √5) / 2))

Explanation:

To find the value of x + 1/x, we first need to solve the equation x^2 + 1/(x^2) = 3.

Let's multiply the equation by x^2 to eliminate the fractions:

x^2(x^2) + 1 = 3x^2

This simplifies to:

x^4 + 1 = 3x^2

Rearranging the equation:

x^4 - 3x^2 + 1 = 0

Now, let's substitute y = x^2:

y^2 - 3y + 1 = 0

This is now a quadratic equation in terms of y. We can solve it using the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / (2a)

For this equation, a = 1, b = -3, and c = 1:

y = (-(-3) ± √((-3)^2 - 4(1)(1))) / (2(1))

Simplifying further:

y = (3 ± √(9 - 4)) / 2

y = (3 ± √5) / 2

Now, let's substitute y back in terms of x:

x^2 = (3 ± √5) / 2

Taking the square root of both sides:

x = ± √((3 ± √5) / 2)

Therefore, the value of x + 1/x can have two possible solutions:

x + 1/x = ± √((3 ± √5) / 2) + 1 / (√((3 ± √5) / 2))

User Pronngo
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