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Simplifying -6n(-6n + -5) = 7n + 30 Reorder the terms: -6n(-5 + -6n) = 7n + 30 (-5 * -6n + -6n * -6n) = 7n + 30 (30n + 36n2) = 7n + 30 Reorder the terms: 30n + 36n2 = 30 + 7n Solving 30n + 36n2 = 30 + 7n Solving for variable 'n'. Reorder the terms: -30 + 30n + -7n + 36n2 = 30 + 7n + -30 + -7n Combine like terms: 30n + -7n = 23n -30 + 23n + 36n2 = 30 + 7n + -30 + -7n Reorder the terms: -30 + 23n + 36n2 = 30 + -30 + 7n + -7n Combine like terms: 30 + -30 = 0 -30 + 23n + 36n2 = 0 + 7n + -7n -30 + 23n + 36n2 = 7n + -7n Combine like terms: 7n + -7n = 0 -30 + 23n + 36n2 = 0 Begin completing the square. Divide all terms by 36 the coefficient of the squared term: Divide each side by '36'. -0.8333333333 + 0.6388888889n + n2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + 0.6388888889n + 0.8333333333 + n2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + 0.6388888889n + n2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + 0.6388888889n + n2 = 0 + 0.8333333333 0.6388888889n + n2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 0.6388888889n + n2 = 0.8333333333 The n term is 0.6388888889n. Take half its coefficient (0.3194444445). Square it (0.1020447531) and add it to both sides. Add '0.1020447531' to each side of the equation. 0.6388888889n + 0.1020447531 + n2 = 0.8333333333 + 0.1020447531 Reorder the terms: 0.1020447531 + 0.6388888889n + n2 = 0.8333333333 + 0.1020447531 Combine like terms: 0.8333333333 + 0.1020447531 = 0.9353780864 0.1020447531 + 0.6388888889n + n2 = 0.9353780864 Factor a perfect square on the left side: (n + 0.3194444445)(n + 0.3194444445) = 0.9353780864 Calculate the square root of the right side: 0.967149464 Break this problem into two subproblems by setting (n + 0.3194444445) equal to 0.967149464 and -0.967149464.Subproblem 1
n + 0.3194444445 = 0.967149464 Simplifying n + 0.3194444445 = 0.967149464 Reorder the terms: 0.3194444445 + n = 0.967149464 Solving 0.3194444445 + n = 0.967149464 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3194444445' to each side of the equation. 0.3194444445 + -0.3194444445 + n = 0.967149464 + -0.3194444445 Combine like terms: 0.3194444445 + -0.3194444445 = 0.0000000000 0.0000000000 + n = 0.967149464 + -0.3194444445 n = 0.967149464 + -0.3194444445 Combine like terms: 0.967149464 + -0.3194444445 = 0.6477050195 n = 0.6477050195 Simplifying n = 0.6477050195Subproblem 2
n + 0.3194444445 = -0.967149464 Simplifying n + 0.3194444445 = -0.967149464 Reorder the terms: 0.3194444445 + n = -0.967149464 Solving 0.3194444445 + n = -0.967149464 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3194444445' to each side of the equation. 0.3194444445 + -0.3194444445 + n = -0.967149464 + -0.3194444445 Combine like terms: 0.3194444445 + -0.3194444445 = 0.0000000000 0.0000000000 + n = -0.967149464 + -0.3194444445 n = -0.967149464 + -0.3194444445 Combine like terms: -0.967149464 + -0.3194444445 = -1.2865939085 n = -1.2865939085 Simplifying n = -1.2865939085Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.6477050195, -1.2865939085}
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