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Solve by completing Square 5n²-6n-2=0​

1 Answer

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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

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