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Simplifying 3r2 + 12r = 23 Reorder the terms: 12r + 3r2 = 23 Solving 12r + 3r2 = 23 Solving for variable 'r'. Reorder the terms: -23 + 12r + 3r2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + 12r + 3r2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -7.666666667 + 4r + r2 = 0 Move the constant term to the right: Add '7.666666667' to each side of the equation. -7.666666667 + 4r + 7.666666667 + r2 = 0 + 7.666666667 Reorder the terms: -7.666666667 + 7.666666667 + 4r + r2 = 0 + 7.666666667 Combine like terms: -7.666666667 + 7.666666667 = 0.000000000 0.000000000 + 4r + r2 = 0 + 7.666666667 4r + r2 = 0 + 7.666666667 Combine like terms: 0 + 7.666666667 = 7.666666667 4r + r2 = 7.666666667 The r term is 4r. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4r + 4 + r2 = 7.666666667 + 4 Reorder the terms: 4 + 4r + r2 = 7.666666667 + 4 Combine like terms: 7.666666667 + 4 = 11.666666667 4 + 4r + r2 = 11.666666667 Factor a perfect square on the left side: (r + 2)(r + 2) = 11.666666667 Calculate the square root of the right side: 3.415650255 Break this problem into two subproblems by setting (r + 2) equal to 3.415650255 and -3.415650255.Subproblem 1
r + 2 = 3.415650255 Simplifying r + 2 = 3.415650255 Reorder the terms: 2 + r = 3.415650255 Solving 2 + r = 3.415650255 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + r = 3.415650255 + -2 Combine like terms: 2 + -2 = 0 0 + r = 3.415650255 + -2 r = 3.415650255 + -2 Combine like terms: 3.415650255 + -2 = 1.415650255 r = 1.415650255 Simplifying r = 1.415650255Subproblem 2
r + 2 = -3.415650255 Simplifying r + 2 = -3.415650255 Reorder the terms: 2 + r = -3.415650255 Solving 2 + r = -3.415650255 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + r = -3.415650255 + -2 Combine like terms: 2 + -2 = 0 0 + r = -3.415650255 + -2 r = -3.415650255 + -2 Combine like terms: -3.415650255 + -2 = -5.415650255 r = -5.415650255 Simplifying r = -5.415650255Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.415650255, -5.415650255}
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