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Simplifying 7r2 + 14r + -40 = 7 Reorder the terms: -40 + 14r + 7r2 = 7 Solving -40 + 14r + 7r2 = 7 Solving for variable 'r'. Reorder the terms: -40 + -7 + 14r + 7r2 = 7 + -7 Combine like terms: -40 + -7 = -47 -47 + 14r + 7r2 = 7 + -7 Combine like terms: 7 + -7 = 0 -47 + 14r + 7r2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -6.714285714 + 2r + r2 = 0 Move the constant term to the right: Add '6.714285714' to each side of the equation. -6.714285714 + 2r + 6.714285714 + r2 = 0 + 6.714285714 Reorder the terms: -6.714285714 + 6.714285714 + 2r + r2 = 0 + 6.714285714 Combine like terms: -6.714285714 + 6.714285714 = 0.000000000 0.000000000 + 2r + r2 = 0 + 6.714285714 2r + r2 = 0 + 6.714285714 Combine like terms: 0 + 6.714285714 = 6.714285714 2r + r2 = 6.714285714 The r term is 2r. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2r + 1 + r2 = 6.714285714 + 1 Reorder the terms: 1 + 2r + r2 = 6.714285714 + 1 Combine like terms: 6.714285714 + 1 = 7.714285714 1 + 2r + r2 = 7.714285714 Factor a perfect square on the left side: (r + 1)(r + 1) = 7.714285714 Calculate the square root of the right side: 2.777460299 Break this problem into two subproblems by setting (r + 1) equal to 2.777460299 and -2.777460299.Subproblem 1
r + 1 = 2.777460299 Simplifying r + 1 = 2.777460299 Reorder the terms: 1 + r = 2.777460299 Solving 1 + r = 2.777460299 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = 2.777460299 + -1 Combine like terms: 1 + -1 = 0 0 + r = 2.777460299 + -1 r = 2.777460299 + -1 Combine like terms: 2.777460299 + -1 = 1.777460299 r = 1.777460299 Simplifying r = 1.777460299Subproblem 2
r + 1 = -2.777460299 Simplifying r + 1 = -2.777460299 Reorder the terms: 1 + r = -2.777460299 Solving 1 + r = -2.777460299 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = -2.777460299 + -1 Combine like terms: 1 + -1 = 0 0 + r = -2.777460299 + -1 r = -2.777460299 + -1 Combine like terms: -2.777460299 + -1 = -3.777460299 r = -3.777460299 Simplifying r = -3.777460299Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.777460299, -3.777460299}
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