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Simplifying 4n2 + -5n + -8 = 12 Reorder the terms: -8 + -5n + 4n2 = 12 Solving -8 + -5n + 4n2 = 12 Solving for variable 'n'. Reorder the terms: -8 + -12 + -5n + 4n2 = 12 + -12 Combine like terms: -8 + -12 = -20 -20 + -5n + 4n2 = 12 + -12 Combine like terms: 12 + -12 = 0 -20 + -5n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5 + -1.25n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -1.25n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -1.25n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -1.25n + n2 = 0 + 5 -1.25n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -1.25n + n2 = 5 The n term is -1.25n. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25n + 0.390625 + n2 = 5 + 0.390625 Reorder the terms: 0.390625 + -1.25n + n2 = 5 + 0.390625 Combine like terms: 5 + 0.390625 = 5.390625 0.390625 + -1.25n + n2 = 5.390625 Factor a perfect square on the left side: (n + -0.625)(n + -0.625) = 5.390625 Calculate the square root of the right side: 2.321771953 Break this problem into two subproblems by setting (n + -0.625) equal to 2.321771953 and -2.321771953.Subproblem 1
n + -0.625 = 2.321771953 Simplifying n + -0.625 = 2.321771953 Reorder the terms: -0.625 + n = 2.321771953 Solving -0.625 + n = 2.321771953 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = 2.321771953 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = 2.321771953 + 0.625 n = 2.321771953 + 0.625 Combine like terms: 2.321771953 + 0.625 = 2.946771953 n = 2.946771953 Simplifying n = 2.946771953Subproblem 2
n + -0.625 = -2.321771953 Simplifying n + -0.625 = -2.321771953 Reorder the terms: -0.625 + n = -2.321771953 Solving -0.625 + n = -2.321771953 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = -2.321771953 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = -2.321771953 + 0.625 n = -2.321771953 + 0.625 Combine like terms: -2.321771953 + 0.625 = -1.696771953 n = -1.696771953 Simplifying n = -1.696771953Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.946771953, -1.696771953}
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