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Simplifying 40r2 + 54r + -7 = 0 Reorder the terms: -7 + 54r + 40r2 = 0 Solving -7 + 54r + 40r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 40 the coefficient of the squared term: Divide each side by '40'. -0.175 + 1.35r + r2 = 0 Move the constant term to the right: Add '0.175' to each side of the equation. -0.175 + 1.35r + 0.175 + r2 = 0 + 0.175 Reorder the terms: -0.175 + 0.175 + 1.35r + r2 = 0 + 0.175 Combine like terms: -0.175 + 0.175 = 0.000 0.000 + 1.35r + r2 = 0 + 0.175 1.35r + r2 = 0 + 0.175 Combine like terms: 0 + 0.175 = 0.175 1.35r + r2 = 0.175 The r term is 1.35r. Take half its coefficient (0.675). Square it (0.455625) and add it to both sides. Add '0.455625' to each side of the equation. 1.35r + 0.455625 + r2 = 0.175 + 0.455625 Reorder the terms: 0.455625 + 1.35r + r2 = 0.175 + 0.455625 Combine like terms: 0.175 + 0.455625 = 0.630625 0.455625 + 1.35r + r2 = 0.630625 Factor a perfect square on the left side: (r + 0.675)(r + 0.675) = 0.630625 Calculate the square root of the right side: 0.794119009 Break this problem into two subproblems by setting (r + 0.675) equal to 0.794119009 and -0.794119009.Subproblem 1
r + 0.675 = 0.794119009 Simplifying r + 0.675 = 0.794119009 Reorder the terms: 0.675 + r = 0.794119009 Solving 0.675 + r = 0.794119009 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.675' to each side of the equation. 0.675 + -0.675 + r = 0.794119009 + -0.675 Combine like terms: 0.675 + -0.675 = 0.000 0.000 + r = 0.794119009 + -0.675 r = 0.794119009 + -0.675 Combine like terms: 0.794119009 + -0.675 = 0.119119009 r = 0.119119009 Simplifying r = 0.119119009Subproblem 2
r + 0.675 = -0.794119009 Simplifying r + 0.675 = -0.794119009 Reorder the terms: 0.675 + r = -0.794119009 Solving 0.675 + r = -0.794119009 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.675' to each side of the equation. 0.675 + -0.675 + r = -0.794119009 + -0.675 Combine like terms: 0.675 + -0.675 = 0.000 0.000 + r = -0.794119009 + -0.675 r = -0.794119009 + -0.675 Combine like terms: -0.794119009 + -0.675 = -1.469119009 r = -1.469119009 Simplifying r = -1.469119009Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.119119009, -1.469119009}
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