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Simplifying 5n2 + -22 = -3n Reorder the terms: -22 + 5n2 = -3n Solving -22 + 5n2 = -3n Solving for variable 'n'. Reorder the terms: -22 + 3n + 5n2 = -3n + 3n Combine like terms: -3n + 3n = 0 -22 + 3n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -4.4 + 0.6n + n2 = 0 Move the constant term to the right: Add '4.4' to each side of the equation. -4.4 + 0.6n + 4.4 + n2 = 0 + 4.4 Reorder the terms: -4.4 + 4.4 + 0.6n + n2 = 0 + 4.4 Combine like terms: -4.4 + 4.4 = 0.0 0.0 + 0.6n + n2 = 0 + 4.4 0.6n + n2 = 0 + 4.4 Combine like terms: 0 + 4.4 = 4.4 0.6n + n2 = 4.4 The n term is 0.6n. Take half its coefficient (0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. 0.6n + 0.09 + n2 = 4.4 + 0.09 Reorder the terms: 0.09 + 0.6n + n2 = 4.4 + 0.09 Combine like terms: 4.4 + 0.09 = 4.49 0.09 + 0.6n + n2 = 4.49 Factor a perfect square on the left side: (n + 0.3)(n + 0.3) = 4.49 Calculate the square root of the right side: 2.11896201 Break this problem into two subproblems by setting (n + 0.3) equal to 2.11896201 and -2.11896201.Subproblem 1
n + 0.3 = 2.11896201 Simplifying n + 0.3 = 2.11896201 Reorder the terms: 0.3 + n = 2.11896201 Solving 0.3 + n = 2.11896201 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + n = 2.11896201 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + n = 2.11896201 + -0.3 n = 2.11896201 + -0.3 Combine like terms: 2.11896201 + -0.3 = 1.81896201 n = 1.81896201 Simplifying n = 1.81896201Subproblem 2
n + 0.3 = -2.11896201 Simplifying n + 0.3 = -2.11896201 Reorder the terms: 0.3 + n = -2.11896201 Solving 0.3 + n = -2.11896201 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + n = -2.11896201 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + n = -2.11896201 + -0.3 n = -2.11896201 + -0.3 Combine like terms: -2.11896201 + -0.3 = -2.41896201 n = -2.41896201 Simplifying n = -2.41896201Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.81896201, -2.41896201}
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