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Simplifying 8n2 + -12 = 3n Reorder the terms: -12 + 8n2 = 3n Solving -12 + 8n2 = 3n Solving for variable 'n'. Reorder the terms: -12 + -3n + 8n2 = 3n + -3n Combine like terms: 3n + -3n = 0 -12 + -3n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.5 + -0.375n + n2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -0.375n + 1.5 + n2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -0.375n + n2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -0.375n + n2 = 0 + 1.5 -0.375n + n2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -0.375n + n2 = 1.5 The n term is -0.375n. Take half its coefficient (-0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. -0.375n + 0.03515625 + n2 = 1.5 + 0.03515625 Reorder the terms: 0.03515625 + -0.375n + n2 = 1.5 + 0.03515625 Combine like terms: 1.5 + 0.03515625 = 1.53515625 0.03515625 + -0.375n + n2 = 1.53515625 Factor a perfect square on the left side: (n + -0.1875)(n + -0.1875) = 1.53515625 Calculate the square root of the right side: 1.239014225 Break this problem into two subproblems by setting (n + -0.1875) equal to 1.239014225 and -1.239014225.Subproblem 1
n + -0.1875 = 1.239014225 Simplifying n + -0.1875 = 1.239014225 Reorder the terms: -0.1875 + n = 1.239014225 Solving -0.1875 + n = 1.239014225 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + n = 1.239014225 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = 1.239014225 + 0.1875 n = 1.239014225 + 0.1875 Combine like terms: 1.239014225 + 0.1875 = 1.426514225 n = 1.426514225 Simplifying n = 1.426514225Subproblem 2
n + -0.1875 = -1.239014225 Simplifying n + -0.1875 = -1.239014225 Reorder the terms: -0.1875 + n = -1.239014225 Solving -0.1875 + n = -1.239014225 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + n = -1.239014225 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = -1.239014225 + 0.1875 n = -1.239014225 + 0.1875 Combine like terms: -1.239014225 + 0.1875 = -1.051514225 n = -1.051514225 Simplifying n = -1.051514225Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.426514225, -1.051514225}
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