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Simplifying 426 = 4n2 + 6n Reorder the terms: 426 = 6n + 4n2 Solving 426 = 6n + 4n2 Solving for variable 'n'. Reorder the terms: 426 + -6n + -4n2 = 6n + -6n + 4n2 + -4n2 Combine like terms: 6n + -6n = 0 426 + -6n + -4n2 = 0 + 4n2 + -4n2 426 + -6n + -4n2 = 4n2 + -4n2 Combine like terms: 4n2 + -4n2 = 0 426 + -6n + -4n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(213 + -3n + -2n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(213 + -3n + -2n2)' equal to zero and attempt to solve: Simplifying 213 + -3n + -2n2 = 0 Solving 213 + -3n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -106.5 + 1.5n + n2 = 0 Move the constant term to the right: Add '106.5' to each side of the equation. -106.5 + 1.5n + 106.5 + n2 = 0 + 106.5 Reorder the terms: -106.5 + 106.5 + 1.5n + n2 = 0 + 106.5 Combine like terms: -106.5 + 106.5 = 0.0 0.0 + 1.5n + n2 = 0 + 106.5 1.5n + n2 = 0 + 106.5 Combine like terms: 0 + 106.5 = 106.5 1.5n + n2 = 106.5 The n term is 1.5n. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5n + 0.5625 + n2 = 106.5 + 0.5625 Reorder the terms: 0.5625 + 1.5n + n2 = 106.5 + 0.5625 Combine like terms: 106.5 + 0.5625 = 107.0625 0.5625 + 1.5n + n2 = 107.0625 Factor a perfect square on the left side: (n + 0.75)(n + 0.75) = 107.0625 Calculate the square root of the right side: 10.347101043 Break this problem into two subproblems by setting (n + 0.75) equal to 10.347101043 and -10.347101043.Subproblem 1
n + 0.75 = 10.347101043 Simplifying n + 0.75 = 10.347101043 Reorder the terms: 0.75 + n = 10.347101043 Solving 0.75 + n = 10.347101043 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = 10.347101043 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 10.347101043 + -0.75 n = 10.347101043 + -0.75 Combine like terms: 10.347101043 + -0.75 = 9.597101043 n = 9.597101043 Simplifying n = 9.597101043Subproblem 2
n + 0.75 = -10.347101043 Simplifying n + 0.75 = -10.347101043 Reorder the terms: 0.75 + n = -10.347101043 Solving 0.75 + n = -10.347101043 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = -10.347101043 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -10.347101043 + -0.75 n = -10.347101043 + -0.75 Combine like terms: -10.347101043 + -0.75 = -11.097101043 n = -11.097101043 Simplifying n = -11.097101043Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.597101043, -11.097101043}Solution
n = {9.597101043, -11.097101043}
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