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Simplifying 5r2 + 7r + -5 = 12 Reorder the terms: -5 + 7r + 5r2 = 12 Solving -5 + 7r + 5r2 = 12 Solving for variable 'r'. Reorder the terms: -5 + -12 + 7r + 5r2 = 12 + -12 Combine like terms: -5 + -12 = -17 -17 + 7r + 5r2 = 12 + -12 Combine like terms: 12 + -12 = 0 -17 + 7r + 5r2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -3.4 + 1.4r + r2 = 0 Move the constant term to the right: Add '3.4' to each side of the equation. -3.4 + 1.4r + 3.4 + r2 = 0 + 3.4 Reorder the terms: -3.4 + 3.4 + 1.4r + r2 = 0 + 3.4 Combine like terms: -3.4 + 3.4 = 0.0 0.0 + 1.4r + r2 = 0 + 3.4 1.4r + r2 = 0 + 3.4 Combine like terms: 0 + 3.4 = 3.4 1.4r + r2 = 3.4 The r term is 1.4r. Take half its coefficient (0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. 1.4r + 0.49 + r2 = 3.4 + 0.49 Reorder the terms: 0.49 + 1.4r + r2 = 3.4 + 0.49 Combine like terms: 3.4 + 0.49 = 3.89 0.49 + 1.4r + r2 = 3.89 Factor a perfect square on the left side: (r + 0.7)(r + 0.7) = 3.89 Calculate the square root of the right side: 1.972308292 Break this problem into two subproblems by setting (r + 0.7) equal to 1.972308292 and -1.972308292.Subproblem 1
r + 0.7 = 1.972308292 Simplifying r + 0.7 = 1.972308292 Reorder the terms: 0.7 + r = 1.972308292 Solving 0.7 + r = 1.972308292 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + r = 1.972308292 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + r = 1.972308292 + -0.7 r = 1.972308292 + -0.7 Combine like terms: 1.972308292 + -0.7 = 1.272308292 r = 1.272308292 Simplifying r = 1.272308292Subproblem 2
r + 0.7 = -1.972308292 Simplifying r + 0.7 = -1.972308292 Reorder the terms: 0.7 + r = -1.972308292 Solving 0.7 + r = -1.972308292 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + r = -1.972308292 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + r = -1.972308292 + -0.7 r = -1.972308292 + -0.7 Combine like terms: -1.972308292 + -0.7 = -2.672308292 r = -2.672308292 Simplifying r = -2.672308292Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.272308292, -2.672308292}
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