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Simplifying 6r2 + 4r + -17 = 7 Reorder the terms: -17 + 4r + 6r2 = 7 Solving -17 + 4r + 6r2 = 7 Solving for variable 'r'. Reorder the terms: -17 + -7 + 4r + 6r2 = 7 + -7 Combine like terms: -17 + -7 = -24 -24 + 4r + 6r2 = 7 + -7 Combine like terms: 7 + -7 = 0 -24 + 4r + 6r2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + 2r + 3r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + 2r + 3r2)' equal to zero and attempt to solve: Simplifying -12 + 2r + 3r2 = 0 Solving -12 + 2r + 3r2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -4 + 0.6666666667r + r2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 0.6666666667r + 4 + r2 = 0 + 4 Reorder the terms: -4 + 4 + 0.6666666667r + r2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 0.6666666667r + r2 = 0 + 4 0.6666666667r + r2 = 0 + 4 Combine like terms: 0 + 4 = 4 0.6666666667r + r2 = 4 The r term is 0.6666666667r. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667r + 0.1111111112 + r2 = 4 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667r + r2 = 4 + 0.1111111112 Combine like terms: 4 + 0.1111111112 = 4.1111111112 0.1111111112 + 0.6666666667r + r2 = 4.1111111112 Factor a perfect square on the left side: (r + 0.3333333334)(r + 0.3333333334) = 4.1111111112 Calculate the square root of the right side: 2.02758751 Break this problem into two subproblems by setting (r + 0.3333333334) equal to 2.02758751 and -2.02758751.Subproblem 1
r + 0.3333333334 = 2.02758751 Simplifying r + 0.3333333334 = 2.02758751 Reorder the terms: 0.3333333334 + r = 2.02758751 Solving 0.3333333334 + r = 2.02758751 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + r = 2.02758751 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + r = 2.02758751 + -0.3333333334 r = 2.02758751 + -0.3333333334 Combine like terms: 2.02758751 + -0.3333333334 = 1.6942541766 r = 1.6942541766 Simplifying r = 1.6942541766Subproblem 2
r + 0.3333333334 = -2.02758751 Simplifying r + 0.3333333334 = -2.02758751 Reorder the terms: 0.3333333334 + r = -2.02758751 Solving 0.3333333334 + r = -2.02758751 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + r = -2.02758751 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + r = -2.02758751 + -0.3333333334 r = -2.02758751 + -0.3333333334 Combine like terms: -2.02758751 + -0.3333333334 = -2.3609208434 r = -2.3609208434 Simplifying r = -2.3609208434Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.6942541766, -2.3609208434}Solution
r = {1.6942541766, -2.3609208434}
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