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Simplifying 5n2 + 9n + -10 = 0 Reorder the terms: -10 + 9n + 5n2 = 0 Solving -10 + 9n + 5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2 + 1.8n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.8n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + 1.8n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.8n + n2 = 0 + 2 1.8n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.8n + n2 = 2 The n term is 1.8n. Take half its coefficient (0.9). Square it (0.81) and add it to both sides. Add '0.81' to each side of the equation. 1.8n + 0.81 + n2 = 2 + 0.81 Reorder the terms: 0.81 + 1.8n + n2 = 2 + 0.81 Combine like terms: 2 + 0.81 = 2.81 0.81 + 1.8n + n2 = 2.81 Factor a perfect square on the left side: (n + 0.9)(n + 0.9) = 2.81 Calculate the square root of the right side: 1.676305461 Break this problem into two subproblems by setting (n + 0.9) equal to 1.676305461 and -1.676305461.Subproblem 1
n + 0.9 = 1.676305461 Simplifying n + 0.9 = 1.676305461 Reorder the terms: 0.9 + n = 1.676305461 Solving 0.9 + n = 1.676305461 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + n = 1.676305461 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + n = 1.676305461 + -0.9 n = 1.676305461 + -0.9 Combine like terms: 1.676305461 + -0.9 = 0.776305461 n = 0.776305461 Simplifying n = 0.776305461Subproblem 2
n + 0.9 = -1.676305461 Simplifying n + 0.9 = -1.676305461 Reorder the terms: 0.9 + n = -1.676305461 Solving 0.9 + n = -1.676305461 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + n = -1.676305461 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + n = -1.676305461 + -0.9 n = -1.676305461 + -0.9 Combine like terms: -1.676305461 + -0.9 = -2.576305461 n = -2.576305461 Simplifying n = -2.576305461Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.776305461, -2.576305461}
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