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Simplifying 14r2 + -48r + -224 = 0 Reorder the terms: -224 + -48r + 14r2 = 0 Solving -224 + -48r + 14r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(-112 + -24r + 7r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-112 + -24r + 7r2)' equal to zero and attempt to solve: Simplifying -112 + -24r + 7r2 = 0 Solving -112 + -24r + 7r2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -16 + -3.428571429r + r2 = 0 Move the constant term to the right: Add '16' to each side of the equation. -16 + -3.428571429r + 16 + r2 = 0 + 16 Reorder the terms: -16 + 16 + -3.428571429r + r2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + -3.428571429r + r2 = 0 + 16 -3.428571429r + r2 = 0 + 16 Combine like terms: 0 + 16 = 16 -3.428571429r + r2 = 16 The r term is -3.428571429r. Take half its coefficient (-1.714285715). Square it (2.938775513) and add it to both sides. Add '2.938775513' to each side of the equation. -3.428571429r + 2.938775513 + r2 = 16 + 2.938775513 Reorder the terms: 2.938775513 + -3.428571429r + r2 = 16 + 2.938775513 Combine like terms: 16 + 2.938775513 = 18.938775513 2.938775513 + -3.428571429r + r2 = 18.938775513 Factor a perfect square on the left side: (r + -1.714285715)(r + -1.714285715) = 18.938775513 Calculate the square root of the right side: 4.351870347 Break this problem into two subproblems by setting (r + -1.714285715) equal to 4.351870347 and -4.351870347.Subproblem 1
r + -1.714285715 = 4.351870347 Simplifying r + -1.714285715 = 4.351870347 Reorder the terms: -1.714285715 + r = 4.351870347 Solving -1.714285715 + r = 4.351870347 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.714285715' to each side of the equation. -1.714285715 + 1.714285715 + r = 4.351870347 + 1.714285715 Combine like terms: -1.714285715 + 1.714285715 = 0.000000000 0.000000000 + r = 4.351870347 + 1.714285715 r = 4.351870347 + 1.714285715 Combine like terms: 4.351870347 + 1.714285715 = 6.066156062 r = 6.066156062 Simplifying r = 6.066156062Subproblem 2
r + -1.714285715 = -4.351870347 Simplifying r + -1.714285715 = -4.351870347 Reorder the terms: -1.714285715 + r = -4.351870347 Solving -1.714285715 + r = -4.351870347 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.714285715' to each side of the equation. -1.714285715 + 1.714285715 + r = -4.351870347 + 1.714285715 Combine like terms: -1.714285715 + 1.714285715 = 0.000000000 0.000000000 + r = -4.351870347 + 1.714285715 r = -4.351870347 + 1.714285715 Combine like terms: -4.351870347 + 1.714285715 = -2.637584632 r = -2.637584632 Simplifying r = -2.637584632Solution
The solution to the problem is based on the solutions from the subproblems. r = {6.066156062, -2.637584632}Solution
r = {6.066156062, -2.637584632}
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