10n+3(8+8n)=6n(n-4)

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Solution for 10n+3(8+8n)=6n(n-4) equation:


Simplifying
10n + 3(8 + 8n) = 6n(n + -4)
10n + (8 * 3 + 8n * 3) = 6n(n + -4)
10n + (24 + 24n) = 6n(n + -4)

Reorder the terms:
24 + 10n + 24n = 6n(n + -4)

Combine like terms: 10n + 24n = 34n
24 + 34n = 6n(n + -4)

Reorder the terms:
24 + 34n = 6n(-4 + n)
24 + 34n = (-4 * 6n + n * 6n)
24 + 34n = (-24n + 6n2)

Solving
24 + 34n = -24n + 6n2

Solving for variable 'n'.

Combine like terms: 34n + 24n = 58n
24 + 58n + -6n2 = -24n + 6n2 + 24n + -6n2

Reorder the terms:
24 + 58n + -6n2 = -24n + 24n + 6n2 + -6n2

Combine like terms: -24n + 24n = 0
24 + 58n + -6n2 = 0 + 6n2 + -6n2
24 + 58n + -6n2 = 6n2 + -6n2

Combine like terms: 6n2 + -6n2 = 0
24 + 58n + -6n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(12 + 29n + -3n2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(12 + 29n + -3n2)' equal to zero and attempt to solve: Simplifying 12 + 29n + -3n2 = 0 Solving 12 + 29n + -3n2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -4 + -9.666666667n + n2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -9.666666667n + 4 + n2 = 0 + 4 Reorder the terms: -4 + 4 + -9.666666667n + n2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -9.666666667n + n2 = 0 + 4 -9.666666667n + n2 = 0 + 4 Combine like terms: 0 + 4 = 4 -9.666666667n + n2 = 4 The n term is -9.666666667n. Take half its coefficient (-4.833333334). Square it (23.36111112) and add it to both sides. Add '23.36111112' to each side of the equation. -9.666666667n + 23.36111112 + n2 = 4 + 23.36111112 Reorder the terms: 23.36111112 + -9.666666667n + n2 = 4 + 23.36111112 Combine like terms: 4 + 23.36111112 = 27.36111112 23.36111112 + -9.666666667n + n2 = 27.36111112 Factor a perfect square on the left side: (n + -4.833333334)(n + -4.833333334) = 27.36111112 Calculate the square root of the right side: 5.230784943 Break this problem into two subproblems by setting (n + -4.833333334) equal to 5.230784943 and -5.230784943.

Subproblem 1

n + -4.833333334 = 5.230784943 Simplifying n + -4.833333334 = 5.230784943 Reorder the terms: -4.833333334 + n = 5.230784943 Solving -4.833333334 + n = 5.230784943 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.833333334' to each side of the equation. -4.833333334 + 4.833333334 + n = 5.230784943 + 4.833333334 Combine like terms: -4.833333334 + 4.833333334 = 0.000000000 0.000000000 + n = 5.230784943 + 4.833333334 n = 5.230784943 + 4.833333334 Combine like terms: 5.230784943 + 4.833333334 = 10.064118277 n = 10.064118277 Simplifying n = 10.064118277

Subproblem 2

n + -4.833333334 = -5.230784943 Simplifying n + -4.833333334 = -5.230784943 Reorder the terms: -4.833333334 + n = -5.230784943 Solving -4.833333334 + n = -5.230784943 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.833333334' to each side of the equation. -4.833333334 + 4.833333334 + n = -5.230784943 + 4.833333334 Combine like terms: -4.833333334 + 4.833333334 = 0.000000000 0.000000000 + n = -5.230784943 + 4.833333334 n = -5.230784943 + 4.833333334 Combine like terms: -5.230784943 + 4.833333334 = -0.397451609 n = -0.397451609 Simplifying n = -0.397451609

Solution

The solution to the problem is based on the solutions from the subproblems. n = {10.064118277, -0.397451609}

Solution

n = {10.064118277, -0.397451609}

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