Question

No links, provide full answer

Answer 1

The value of M is:

M = ± (1 / 2) · √[(k² + 5)² /(4 · k²) - 5] ∓ (1 / 2) · √[(k² / 100) · (25 / k² + 5)² - 5] + √3, for k ≠ 0.

What are the family of solutions behind an equation of the form p · q = k?

In this question we have a product of two nonzero real numbers equal to 5, whose mathematical form is:

p · q = 5, where p = 2 · x + √(4 · x² + 5) = k and q = 2 · y + √(4 · y² + 5) = 5 / k

Then, we proceed to find the solutions for x and y in terms of k:

2 · x + √(4 · x² + 5) = k

2 · x = k - √(4 · x² + 5)

4 · x² = k² - 2 · k · √(4 · x² + 5) + 4 · x² + 5

2 · k · √(4 · x² + 5) = k² + 5

4 · k² · (4 · x² + 5) = (k² + 5)²

4 · x² + 5 = (k² + 5)² /(4 · k²)

4 · x² = (k² + 5)² /(4 · k²) - 5

x = ± (1 / 2) · √[(k² + 5)² /(4 · k²) - 5], for k ≠ 0.

2 · y = 5 / k - √(4 · y² + 5)

4 · y² = 25 / k² - (10 / k) · √(4 · y² + 5) + 4 · y² + 5

(10 / k) · √(4 · y² + 5) = 25 / k² + 5

(100 / k²) · (4 · y² + 5) = (25 / k² + 5)²

4 · y² + 5 = (k² / 100) · (25 / k² + 5)²

4 · y² = (k² / 100) · (25 / k² + 5)² - 5

y = ∓ (1 / 2) · √[(k² / 100) · (25 / k² + 5)² - 5], for k ≠ 0.

Therefore, the value of M is:

M = ± (1 / 2) · √[(k² + 5)² /(4 · k²) - 5] ∓ (1 / 2) · √[(k² / 100) · (25 / k² + 5)² - 5] + √3, for k ≠ 0.

To learn more on radical equations: brainly.com/question/8606917

#SPJ1