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Simplifying 10n2 = 6n + 24 Reorder the terms: 10n2 = 24 + 6n Solving 10n2 = 24 + 6n Solving for variable 'n'. Reorder the terms: -24 + -6n + 10n2 = 24 + 6n + -24 + -6n Reorder the terms: -24 + -6n + 10n2 = 24 + -24 + 6n + -6n Combine like terms: 24 + -24 = 0 -24 + -6n + 10n2 = 0 + 6n + -6n -24 + -6n + 10n2 = 6n + -6n Combine like terms: 6n + -6n = 0 -24 + -6n + 10n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + -3n + 5n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + -3n + 5n2)' equal to zero and attempt to solve: Simplifying -12 + -3n + 5n2 = 0 Solving -12 + -3n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.4 + -0.6n + n2 = 0 Move the constant term to the right: Add '2.4' to each side of the equation. -2.4 + -0.6n + 2.4 + n2 = 0 + 2.4 Reorder the terms: -2.4 + 2.4 + -0.6n + n2 = 0 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + -0.6n + n2 = 0 + 2.4 -0.6n + n2 = 0 + 2.4 Combine like terms: 0 + 2.4 = 2.4 -0.6n + n2 = 2.4 The n term is -0.6n. Take half its coefficient (-0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. -0.6n + 0.09 + n2 = 2.4 + 0.09 Reorder the terms: 0.09 + -0.6n + n2 = 2.4 + 0.09 Combine like terms: 2.4 + 0.09 = 2.49 0.09 + -0.6n + n2 = 2.49 Factor a perfect square on the left side: (n + -0.3)(n + -0.3) = 2.49 Calculate the square root of the right side: 1.577973384 Break this problem into two subproblems by setting (n + -0.3) equal to 1.577973384 and -1.577973384.Subproblem 1
n + -0.3 = 1.577973384 Simplifying n + -0.3 = 1.577973384 Reorder the terms: -0.3 + n = 1.577973384 Solving -0.3 + n = 1.577973384 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + n = 1.577973384 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = 1.577973384 + 0.3 n = 1.577973384 + 0.3 Combine like terms: 1.577973384 + 0.3 = 1.877973384 n = 1.877973384 Simplifying n = 1.877973384Subproblem 2
n + -0.3 = -1.577973384 Simplifying n + -0.3 = -1.577973384 Reorder the terms: -0.3 + n = -1.577973384 Solving -0.3 + n = -1.577973384 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + n = -1.577973384 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = -1.577973384 + 0.3 n = -1.577973384 + 0.3 Combine like terms: -1.577973384 + 0.3 = -1.277973384 n = -1.277973384 Simplifying n = -1.277973384Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.877973384, -1.277973384}Solution
n = {1.877973384, -1.277973384}
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