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Simplifying 4r2 + -1 = 7r Reorder the terms: -1 + 4r2 = 7r Solving -1 + 4r2 = 7r Solving for variable 'r'. Reorder the terms: -1 + -7r + 4r2 = 7r + -7r Combine like terms: 7r + -7r = 0 -1 + -7r + 4r2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + -1.75r + r2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1.75r + 0.25 + r2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1.75r + r2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1.75r + r2 = 0 + 0.25 -1.75r + r2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1.75r + r2 = 0.25 The r term is -1.75r. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75r + 0.765625 + r2 = 0.25 + 0.765625 Reorder the terms: 0.765625 + -1.75r + r2 = 0.25 + 0.765625 Combine like terms: 0.25 + 0.765625 = 1.015625 0.765625 + -1.75r + r2 = 1.015625 Factor a perfect square on the left side: (r + -0.875)(r + -0.875) = 1.015625 Calculate the square root of the right side: 1.007782219 Break this problem into two subproblems by setting (r + -0.875) equal to 1.007782219 and -1.007782219.Subproblem 1
r + -0.875 = 1.007782219 Simplifying r + -0.875 = 1.007782219 Reorder the terms: -0.875 + r = 1.007782219 Solving -0.875 + r = 1.007782219 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + r = 1.007782219 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + r = 1.007782219 + 0.875 r = 1.007782219 + 0.875 Combine like terms: 1.007782219 + 0.875 = 1.882782219 r = 1.882782219 Simplifying r = 1.882782219Subproblem 2
r + -0.875 = -1.007782219 Simplifying r + -0.875 = -1.007782219 Reorder the terms: -0.875 + r = -1.007782219 Solving -0.875 + r = -1.007782219 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + r = -1.007782219 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + r = -1.007782219 + 0.875 r = -1.007782219 + 0.875 Combine like terms: -1.007782219 + 0.875 = -0.132782219 r = -0.132782219 Simplifying r = -0.132782219Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.882782219, -0.132782219}
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