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Simplifying 4r2 = -2r + 14 Reorder the terms: 4r2 = 14 + -2r Solving 4r2 = 14 + -2r Solving for variable 'r'. Reorder the terms: -14 + 2r + 4r2 = 14 + -2r + -14 + 2r Reorder the terms: -14 + 2r + 4r2 = 14 + -14 + -2r + 2r Combine like terms: 14 + -14 = 0 -14 + 2r + 4r2 = 0 + -2r + 2r -14 + 2r + 4r2 = -2r + 2r Combine like terms: -2r + 2r = 0 -14 + 2r + 4r2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + r + 2r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + r + 2r2)' equal to zero and attempt to solve: Simplifying -7 + r + 2r2 = 0 Solving -7 + r + 2r2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + 0.5r + r2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 0.5r + 3.5 + r2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 0.5r + r2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 0.5r + r2 = 0 + 3.5 0.5r + r2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 0.5r + r2 = 3.5 The r term is r. 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. 0.5r + 0.25 + r2 = 3.5 + 0.25 Reorder the terms: 0.25 + 0.5r + r2 = 3.5 + 0.25 Combine like terms: 3.5 + 0.25 = 3.75 0.25 + 0.5r + r2 = 3.75 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (r + 0.5) equal to 1.936491673 and -1.936491673.Subproblem 1
r + 0.5 = 1.936491673 Simplifying r + 0.5 = 1.936491673 Reorder the terms: 0.5 + r = 1.936491673 Solving 0.5 + r = 1.936491673 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 = 1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 1.936491673 + -0.5 r = 1.936491673 + -0.5 Combine like terms: 1.936491673 + -0.5 = 1.436491673 r = 1.436491673 Simplifying r = 1.436491673Subproblem 2
r + 0.5 = -1.936491673 Simplifying r + 0.5 = -1.936491673 Reorder the terms: 0.5 + r = -1.936491673 Solving 0.5 + r = -1.936491673 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 = -1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -1.936491673 + -0.5 r = -1.936491673 + -0.5 Combine like terms: -1.936491673 + -0.5 = -2.436491673 r = -2.436491673 Simplifying r = -2.436491673Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.436491673, -2.436491673}Solution
r = {1.436491673, -2.436491673}
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