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Simplifying 16 = 3.14r2 + 2r(4) Reorder the terms for easier multiplication: 16 = 3.14r2 + 2 * 4r Multiply 2 * 4 16 = 3.14r2 + 8r Reorder the terms: 16 = 8r + 3.14r2 Solving 16 = 8r + 3.14r2 Solving for variable 'r'. Reorder the terms: 16 + -8r + -3.14r2 = 8r + -8r + 3.14r2 + -3.14r2 Combine like terms: 8r + -8r = 0 16 + -8r + -3.14r2 = 0 + 3.14r2 + -3.14r2 16 + -8r + -3.14r2 = 3.14r2 + -3.14r2 Combine like terms: 3.14r2 + -3.14r2 = 0.00 16 + -8r + -3.14r2 = 0.00 Begin completing the square. Divide all terms by -3.14 the coefficient of the squared term: Divide each side by '-3.14'. -5.095541401 + 2.547770701r + r2 = 0 Move the constant term to the right: Add '5.095541401' to each side of the equation. -5.095541401 + 2.547770701r + 5.095541401 + r2 = 0 + 5.095541401 Reorder the terms: -5.095541401 + 5.095541401 + 2.547770701r + r2 = 0 + 5.095541401 Combine like terms: -5.095541401 + 5.095541401 = 0.000000000 0.000000000 + 2.547770701r + r2 = 0 + 5.095541401 2.547770701r + r2 = 0 + 5.095541401 Combine like terms: 0 + 5.095541401 = 5.095541401 2.547770701r + r2 = 5.095541401 The r term is 2.547770701r. Take half its coefficient (1.273885351). Square it (1.622783887) and add it to both sides. Add '1.622783887' to each side of the equation. 2.547770701r + 1.622783887 + r2 = 5.095541401 + 1.622783887 Reorder the terms: 1.622783887 + 2.547770701r + r2 = 5.095541401 + 1.622783887 Combine like terms: 5.095541401 + 1.622783887 = 6.718325288 1.622783887 + 2.547770701r + r2 = 6.718325288 Factor a perfect square on the left side: (r + 1.273885351)(r + 1.273885351) = 6.718325288 Calculate the square root of the right side: 2.591973242 Break this problem into two subproblems by setting (r + 1.273885351) equal to 2.591973242 and -2.591973242.Subproblem 1
r + 1.273885351 = 2.591973242 Simplifying r + 1.273885351 = 2.591973242 Reorder the terms: 1.273885351 + r = 2.591973242 Solving 1.273885351 + r = 2.591973242 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.273885351' to each side of the equation. 1.273885351 + -1.273885351 + r = 2.591973242 + -1.273885351 Combine like terms: 1.273885351 + -1.273885351 = 0.000000000 0.000000000 + r = 2.591973242 + -1.273885351 r = 2.591973242 + -1.273885351 Combine like terms: 2.591973242 + -1.273885351 = 1.318087891 r = 1.318087891 Simplifying r = 1.318087891Subproblem 2
r + 1.273885351 = -2.591973242 Simplifying r + 1.273885351 = -2.591973242 Reorder the terms: 1.273885351 + r = -2.591973242 Solving 1.273885351 + r = -2.591973242 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.273885351' to each side of the equation. 1.273885351 + -1.273885351 + r = -2.591973242 + -1.273885351 Combine like terms: 1.273885351 + -1.273885351 = 0.000000000 0.000000000 + r = -2.591973242 + -1.273885351 r = -2.591973242 + -1.273885351 Combine like terms: -2.591973242 + -1.273885351 = -3.865858593 r = -3.865858593 Simplifying r = -3.865858593Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.318087891, -3.865858593}
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