6n^2-10n=10

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Solution for 6n^2-10n=10 equation:


Simplifying
6n2 + -10n = 10

Reorder the terms:
-10n + 6n2 = 10

Solving
-10n + 6n2 = 10

Solving for variable 'n'.

Reorder the terms:
-10 + -10n + 6n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + -10n + 6n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-5 + -5n + 3n2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(-5 + -5n + 3n2)' equal to zero and attempt to solve: Simplifying -5 + -5n + 3n2 = 0 Solving -5 + -5n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.666666667 + -1.666666667n + n2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + -1.666666667n + 1.666666667 + n2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + -1.666666667n + n2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + -1.666666667n + n2 = 0 + 1.666666667 -1.666666667n + n2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 -1.666666667n + n2 = 1.666666667 The n term is -1.666666667n. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667n + 0.6944444447 + n2 = 1.666666667 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667n + n2 = 1.666666667 + 0.6944444447 Combine like terms: 1.666666667 + 0.6944444447 = 2.3611111117 0.6944444447 + -1.666666667n + n2 = 2.3611111117 Factor a perfect square on the left side: (n + -0.8333333335)(n + -0.8333333335) = 2.3611111117 Calculate the square root of the right side: 1.536590743 Break this problem into two subproblems by setting (n + -0.8333333335) equal to 1.536590743 and -1.536590743.

Subproblem 1

n + -0.8333333335 = 1.536590743 Simplifying n + -0.8333333335 = 1.536590743 Reorder the terms: -0.8333333335 + n = 1.536590743 Solving -0.8333333335 + n = 1.536590743 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = 1.536590743 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = 1.536590743 + 0.8333333335 n = 1.536590743 + 0.8333333335 Combine like terms: 1.536590743 + 0.8333333335 = 2.3699240765 n = 2.3699240765 Simplifying n = 2.3699240765

Subproblem 2

n + -0.8333333335 = -1.536590743 Simplifying n + -0.8333333335 = -1.536590743 Reorder the terms: -0.8333333335 + n = -1.536590743 Solving -0.8333333335 + n = -1.536590743 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = -1.536590743 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = -1.536590743 + 0.8333333335 n = -1.536590743 + 0.8333333335 Combine like terms: -1.536590743 + 0.8333333335 = -0.7032574095 n = -0.7032574095 Simplifying n = -0.7032574095

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.3699240765, -0.7032574095}

Solution

n = {2.3699240765, -0.7032574095}

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