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Simplifying -6r2 + 5r + 3 = 0 Reorder the terms: 3 + 5r + -6r2 = 0 Solving 3 + 5r + -6r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -0.5 + -0.8333333333r + r2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -0.8333333333r + 0.5 + r2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -0.8333333333r + r2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -0.8333333333r + r2 = 0 + 0.5 -0.8333333333r + r2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -0.8333333333r + r2 = 0.5 The r term is -0.8333333333r. Take half its coefficient (-0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. -0.8333333333r + 0.1736111111 + r2 = 0.5 + 0.1736111111 Reorder the terms: 0.1736111111 + -0.8333333333r + r2 = 0.5 + 0.1736111111 Combine like terms: 0.5 + 0.1736111111 = 0.6736111111 0.1736111111 + -0.8333333333r + r2 = 0.6736111111 Factor a perfect square on the left side: (r + -0.4166666667)(r + -0.4166666667) = 0.6736111111 Calculate the square root of the right side: 0.82073815 Break this problem into two subproblems by setting (r + -0.4166666667) equal to 0.82073815 and -0.82073815.Subproblem 1
r + -0.4166666667 = 0.82073815 Simplifying r + -0.4166666667 = 0.82073815 Reorder the terms: -0.4166666667 + r = 0.82073815 Solving -0.4166666667 + r = 0.82073815 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + r = 0.82073815 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + r = 0.82073815 + 0.4166666667 r = 0.82073815 + 0.4166666667 Combine like terms: 0.82073815 + 0.4166666667 = 1.2374048167 r = 1.2374048167 Simplifying r = 1.2374048167Subproblem 2
r + -0.4166666667 = -0.82073815 Simplifying r + -0.4166666667 = -0.82073815 Reorder the terms: -0.4166666667 + r = -0.82073815 Solving -0.4166666667 + r = -0.82073815 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + r = -0.82073815 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + r = -0.82073815 + 0.4166666667 r = -0.82073815 + 0.4166666667 Combine like terms: -0.82073815 + 0.4166666667 = -0.4040714833 r = -0.4040714833 Simplifying r = -0.4040714833Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.2374048167, -0.4040714833}
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