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Simplifying 7r2 + 2r + 4 = 5 Reorder the terms: 4 + 2r + 7r2 = 5 Solving 4 + 2r + 7r2 = 5 Solving for variable 'r'. Reorder the terms: 4 + -5 + 2r + 7r2 = 5 + -5 Combine like terms: 4 + -5 = -1 -1 + 2r + 7r2 = 5 + -5 Combine like terms: 5 + -5 = 0 -1 + 2r + 7r2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.1428571429 + 0.2857142857r + r2 = 0 Move the constant term to the right: Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.2857142857r + 0.1428571429 + r2 = 0 + 0.1428571429 Reorder the terms: -0.1428571429 + 0.1428571429 + 0.2857142857r + r2 = 0 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + 0.2857142857r + r2 = 0 + 0.1428571429 0.2857142857r + r2 = 0 + 0.1428571429 Combine like terms: 0 + 0.1428571429 = 0.1428571429 0.2857142857r + r2 = 0.1428571429 The r term is 0.2857142857r. Take half its coefficient (0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. 0.2857142857r + 0.02040816328 + r2 = 0.1428571429 + 0.02040816328 Reorder the terms: 0.02040816328 + 0.2857142857r + r2 = 0.1428571429 + 0.02040816328 Combine like terms: 0.1428571429 + 0.02040816328 = 0.16326530618 0.02040816328 + 0.2857142857r + r2 = 0.16326530618 Factor a perfect square on the left side: (r + 0.1428571429)(r + 0.1428571429) = 0.16326530618 Calculate the square root of the right side: 0.404061018 Break this problem into two subproblems by setting (r + 0.1428571429) equal to 0.404061018 and -0.404061018.Subproblem 1
r + 0.1428571429 = 0.404061018 Simplifying r + 0.1428571429 = 0.404061018 Reorder the terms: 0.1428571429 + r = 0.404061018 Solving 0.1428571429 + r = 0.404061018 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.1428571429' to each side of the equation. 0.1428571429 + -0.1428571429 + r = 0.404061018 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + r = 0.404061018 + -0.1428571429 r = 0.404061018 + -0.1428571429 Combine like terms: 0.404061018 + -0.1428571429 = 0.2612038751 r = 0.2612038751 Simplifying r = 0.2612038751Subproblem 2
r + 0.1428571429 = -0.404061018 Simplifying r + 0.1428571429 = -0.404061018 Reorder the terms: 0.1428571429 + r = -0.404061018 Solving 0.1428571429 + r = -0.404061018 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.1428571429' to each side of the equation. 0.1428571429 + -0.1428571429 + r = -0.404061018 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + r = -0.404061018 + -0.1428571429 r = -0.404061018 + -0.1428571429 Combine like terms: -0.404061018 + -0.1428571429 = -0.5469181609 r = -0.5469181609 Simplifying r = -0.5469181609Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.2612038751, -0.5469181609}
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