Answer:
n = 3/5 + sqrt(19)/5, or n = 3/5 - sqrt(19)/5
Explanation:
To solve the quadratic equation 5n² - 6n - 2 = 0 by completing the square, we can follow these steps:
Move the constant term to the right-hand side of the equation:
5n² - 6n = 2
Divide both sides by the coefficient of the squared term, 5, to make the coefficient 1:
n² - (6/5)n = 2/5
Take half of the coefficient of the linear term, -6/5, and square it:
(-6/5) / 2 = -3/5
(-3/5)² = 9/25
Add this value to both sides of the equation:
n² - (6/5)n + 9/25 = 2/5 + 9/25
Simplify the right-hand side:
2/5 + 9/25 = 10/25 + 9/25 = 19/25
Factor the left-hand side as a perfect square:
(n - 3/5)² = 19/25
Take the square root of both sides, remembering to include ±:
n - 3/5 = ±sqrt(19)/5
Add 3/5 to both sides:
n = 3/5 ±sqrt(19)/5
Therefore, the solutions to the equation 5n² - 6n - 2 = 0 are:
n = 3/5 + sqrt(19)/5, or n = 3/5 - sqrt(19)/5