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Simplifying 5r2 = 10 + -9r Solving 5r2 = 10 + -9r Solving for variable 'r'. Reorder the terms: -10 + 9r + 5r2 = 10 + -9r + -10 + 9r Reorder the terms: -10 + 9r + 5r2 = 10 + -10 + -9r + 9r Combine like terms: 10 + -10 = 0 -10 + 9r + 5r2 = 0 + -9r + 9r -10 + 9r + 5r2 = -9r + 9r Combine like terms: -9r + 9r = 0 -10 + 9r + 5r2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2 + 1.8r + r2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.8r + 2 + r2 = 0 + 2 Reorder the terms: -2 + 2 + 1.8r + r2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.8r + r2 = 0 + 2 1.8r + r2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.8r + r2 = 2 The r term is 1.8r. 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. 1.8r + 0.81 + r2 = 2 + 0.81 Reorder the terms: 0.81 + 1.8r + r2 = 2 + 0.81 Combine like terms: 2 + 0.81 = 2.81 0.81 + 1.8r + r2 = 2.81 Factor a perfect square on the left side: (r + 0.9)(r + 0.9) = 2.81 Calculate the square root of the right side: 1.676305461 Break this problem into two subproblems by setting (r + 0.9) equal to 1.676305461 and -1.676305461.Subproblem 1
r + 0.9 = 1.676305461 Simplifying r + 0.9 = 1.676305461 Reorder the terms: 0.9 + r = 1.676305461 Solving 0.9 + r = 1.676305461 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + r = 1.676305461 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + r = 1.676305461 + -0.9 r = 1.676305461 + -0.9 Combine like terms: 1.676305461 + -0.9 = 0.776305461 r = 0.776305461 Simplifying r = 0.776305461Subproblem 2
r + 0.9 = -1.676305461 Simplifying r + 0.9 = -1.676305461 Reorder the terms: 0.9 + r = -1.676305461 Solving 0.9 + r = -1.676305461 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + r = -1.676305461 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + r = -1.676305461 + -0.9 r = -1.676305461 + -0.9 Combine like terms: -1.676305461 + -0.9 = -2.576305461 r = -2.576305461 Simplifying r = -2.576305461Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.776305461, -2.576305461}
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