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Simplifying -90 = -5(-1n + 22) * 5n Reorder the terms: -90 = -5(22 + -1n) * 5n Reorder the terms for easier multiplication: -90 = -5 * 5n(22 + -1n) Multiply -5 * 5 -90 = -25n(22 + -1n) -90 = (22 * -25n + -1n * -25n) -90 = (-550n + 25n2) Solving -90 = -550n + 25n2 Solving for variable 'n'. Reorder the terms: -90 + 550n + -25n2 = -550n + 550n + 25n2 + -25n2 Combine like terms: -550n + 550n = 0 -90 + 550n + -25n2 = 0 + 25n2 + -25n2 -90 + 550n + -25n2 = 25n2 + -25n2 Combine like terms: 25n2 + -25n2 = 0 -90 + 550n + -25n2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-18 + 110n + -5n2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-18 + 110n + -5n2)' equal to zero and attempt to solve: Simplifying -18 + 110n + -5n2 = 0 Solving -18 + 110n + -5n2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 3.6 + -22n + n2 = 0 Move the constant term to the right: Add '-3.6' to each side of the equation. 3.6 + -22n + -3.6 + n2 = 0 + -3.6 Reorder the terms: 3.6 + -3.6 + -22n + n2 = 0 + -3.6 Combine like terms: 3.6 + -3.6 = 0.0 0.0 + -22n + n2 = 0 + -3.6 -22n + n2 = 0 + -3.6 Combine like terms: 0 + -3.6 = -3.6 -22n + n2 = -3.6 The n term is -22n. Take half its coefficient (-11). Square it (121) and add it to both sides. Add '121' to each side of the equation. -22n + 121 + n2 = -3.6 + 121 Reorder the terms: 121 + -22n + n2 = -3.6 + 121 Combine like terms: -3.6 + 121 = 117.4 121 + -22n + n2 = 117.4 Factor a perfect square on the left side: (n + -11)(n + -11) = 117.4 Calculate the square root of the right side: 10.835128056 Break this problem into two subproblems by setting (n + -11) equal to 10.835128056 and -10.835128056.Subproblem 1
n + -11 = 10.835128056 Simplifying n + -11 = 10.835128056 Reorder the terms: -11 + n = 10.835128056 Solving -11 + n = 10.835128056 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + n = 10.835128056 + 11 Combine like terms: -11 + 11 = 0 0 + n = 10.835128056 + 11 n = 10.835128056 + 11 Combine like terms: 10.835128056 + 11 = 21.835128056 n = 21.835128056 Simplifying n = 21.835128056Subproblem 2
n + -11 = -10.835128056 Simplifying n + -11 = -10.835128056 Reorder the terms: -11 + n = -10.835128056 Solving -11 + n = -10.835128056 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + n = -10.835128056 + 11 Combine like terms: -11 + 11 = 0 0 + n = -10.835128056 + 11 n = -10.835128056 + 11 Combine like terms: -10.835128056 + 11 = 0.164871944 n = 0.164871944 Simplifying n = 0.164871944Solution
The solution to the problem is based on the solutions from the subproblems. n = {21.835128056, 0.164871944}Solution
n = {21.835128056, 0.164871944}
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