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Simplifying -5r2 + 5r + 4 = 0 Reorder the terms: 4 + 5r + -5r2 = 0 Solving 4 + 5r + -5r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -0.8 + -1r + r2 = 0 Move the constant term to the right: Add '0.8' to each side of the equation. -0.8 + -1r + 0.8 + r2 = 0 + 0.8 Reorder the terms: -0.8 + 0.8 + -1r + r2 = 0 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + -1r + r2 = 0 + 0.8 -1r + r2 = 0 + 0.8 Combine like terms: 0 + 0.8 = 0.8 -1r + r2 = 0.8 The r term is -1r. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1r + 0.25 + r2 = 0.8 + 0.25 Reorder the terms: 0.25 + -1r + r2 = 0.8 + 0.25 Combine like terms: 0.8 + 0.25 = 1.05 0.25 + -1r + r2 = 1.05 Factor a perfect square on the left side: (r + -0.5)(r + -0.5) = 1.05 Calculate the square root of the right side: 1.024695077 Break this problem into two subproblems by setting (r + -0.5) equal to 1.024695077 and -1.024695077.Subproblem 1
r + -0.5 = 1.024695077 Simplifying r + -0.5 = 1.024695077 Reorder the terms: -0.5 + r = 1.024695077 Solving -0.5 + r = 1.024695077 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + r = 1.024695077 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = 1.024695077 + 0.5 r = 1.024695077 + 0.5 Combine like terms: 1.024695077 + 0.5 = 1.524695077 r = 1.524695077 Simplifying r = 1.524695077Subproblem 2
r + -0.5 = -1.024695077 Simplifying r + -0.5 = -1.024695077 Reorder the terms: -0.5 + r = -1.024695077 Solving -0.5 + r = -1.024695077 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + r = -1.024695077 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = -1.024695077 + 0.5 r = -1.024695077 + 0.5 Combine like terms: -1.024695077 + 0.5 = -0.524695077 r = -0.524695077 Simplifying r = -0.524695077Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.524695077, -0.524695077}
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