10-n(3n-4)-n^2=

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Solution for 10-n(3n-4)-n^2= equation:


Simplifying
10 + -1n(3n + -4) + -1n2 = 0

Reorder the terms:
10 + -1n(-4 + 3n) + -1n2 = 0
10 + (-4 * -1n + 3n * -1n) + -1n2 = 0
10 + (4n + -3n2) + -1n2 = 0

Combine like terms: -3n2 + -1n2 = -4n2
10 + 4n + -4n2 = 0

Solving
10 + 4n + -4n2 = 0

Solving for variable 'n'.

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

Ignore the factor 2.

Subproblem 1

Set the factor '(5 + 2n + -2n2)' equal to zero and attempt to solve: Simplifying 5 + 2n + -2n2 = 0 Solving 5 + 2n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + -1n + n2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -1n + 2.5 + n2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -1n + n2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -1n + n2 = 0 + 2.5 -1n + n2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -1n + n2 = 2.5 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 2.5 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 2.5 + 0.25 Combine like terms: 2.5 + 0.25 = 2.75 0.25 + -1n + n2 = 2.75 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (n + -0.5) equal to 1.658312395 and -1.658312395.

Subproblem 1

n + -0.5 = 1.658312395 Simplifying n + -0.5 = 1.658312395 Reorder the terms: -0.5 + n = 1.658312395 Solving -0.5 + n = 1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 1.658312395 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 1.658312395 + 0.5 n = 1.658312395 + 0.5 Combine like terms: 1.658312395 + 0.5 = 2.158312395 n = 2.158312395 Simplifying n = 2.158312395

Subproblem 2

n + -0.5 = -1.658312395 Simplifying n + -0.5 = -1.658312395 Reorder the terms: -0.5 + n = -1.658312395 Solving -0.5 + n = -1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -1.658312395 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -1.658312395 + 0.5 n = -1.658312395 + 0.5 Combine like terms: -1.658312395 + 0.5 = -1.158312395 n = -1.158312395 Simplifying n = -1.158312395

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.158312395, -1.158312395}

Solution

n = {2.158312395, -1.158312395}

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