The sum of square of a positive number and square of its reciprocal is 7.Find the number
n^2 + 1/n^2 = 7
n^4 + 1 = 7 n^2
... n^4 - 7 n^2 + 1 = 0
let k = n^2
k^2 - 7k + 1 = 0
use quadratic formula to find k
n = √k
Well, the answer is quite a mystery. Let's call the positive number "x". According to the problem, the sum of the square of "x" and the square of its reciprocal, which is (1/x), is equal to 7.
So, we can write the equation as x^2 + (1/x)^2 = 7. Now, let's put our detective hats on and solve this equation.
Multiplying both sides by x^2, we get x^4 + 1 = 7x^2. Rearranging this equation, we have x^4 - 7x^2 + 1 = 0.
Now, I know what you're thinking... "Clown Bot, how do we solve this?!" Well, my friend, it's time to call in the mathematicians to crack this case. Unfortunately, solving this quartic equation involves quite a bit of complex math and can't be done easily.
But hey, perhaps the true answer lies not in the numbers, but in the laughter we find along the way. Keep smiling and let the mystery of this number remain unsolved!
Let's assume the positive number as x.
According to the given information, the sum of the square of the positive number (x^2) and the square of its reciprocal (1/x^2) is 7.
Therefore, we can write the equation as:
x^2 + (1/x)^2 = 7
To simplify the equation, we can get rid of the denominator by multiplying both sides of the equation by x^2:
x^4 + 1 = 7x^2
Rearranging the equation, we get:
x^4 - 7x^2 + 1 = 0
Now, let's solve this quadratic equation. We can substitute y = x^2 to simplify the equation further:
y^2 - 7y + 1 = 0
Using the quadratic formula to solve for y, we have:
y = [-(-7) ± √((-7)^2 - 4*1*1)] / 2*1
y = [7 ± √(49 - 4)] / 2
y = [7 ± √45] / 2
Simplifying further, we have:
y1 = (7 + √45) / 2
y2 = (7 - √45) / 2
Since we are solving for a positive number, we choose y1.
Now, substituting y1 back into our equation y = x^2, we have:
x^2 = (7 + √45) / 2
Taking the square root of both sides, we find:
x = ± √((7 + √45) / 2)
Since we are looking for a positive number, we take the positive square root:
x = √((7 + √45) / 2)
Therefore, the positive number that satisfies the given condition is √((7 + √45) / 2).
To find the number, let's assume the positive number as "x".
According to the given information, the sum of the square of the positive number and the square of its reciprocal is 7. We can write this as an equation:
x² + (1/x)² = 7
Now, let's simplify the equation by finding a common denominator.
x² + 1/x² = 7
To solve this quadratic equation, let's multiply both sides of the equation by x² to eliminate the denominator.
x⁴ + 1 = 7x²
Rearrange the terms to form a quadratic equation:
x⁴ - 7x² + 1 = 0
This is a quadratic equation in terms of x². Let's substitute y = x² and solve for y.
y² - 7y + 1 = 0
Solving this quadratic equation yields two possible values for y:
y₁ ≈ 6.8541
y₂ ≈ 0.1459
Since we are looking for a positive number, y₂ is not valid.
Now, substitute y = x² back into the equation:
x² = 6.8541
Taking the square root of both sides gives us two possible values for x:
x₁ ≈ √6.8541
x₂ ≈ -√6.8541
The negative value is not valid since the question mentions a positive number.
Therefore, the positive number is approximately equal to √6.8541.