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: https://brainly.com/question/8606917
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