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Simplifying 125 = (4r + 2)(5r + -12) Reorder the terms: 125 = (2 + 4r)(5r + -12) Reorder the terms: 125 = (2 + 4r)(-12 + 5r) Multiply (2 + 4r) * (-12 + 5r) 125 = (2(-12 + 5r) + 4r * (-12 + 5r)) 125 = ((-12 * 2 + 5r * 2) + 4r * (-12 + 5r)) 125 = ((-24 + 10r) + 4r * (-12 + 5r)) 125 = (-24 + 10r + (-12 * 4r + 5r * 4r)) 125 = (-24 + 10r + (-48r + 20r2)) Combine like terms: 10r + -48r = -38r 125 = (-24 + -38r + 20r2) Solving 125 = -24 + -38r + 20r2 Solving for variable 'r'. Combine like terms: 125 + 24 = 149 149 + 38r + -20r2 = -24 + -38r + 20r2 + 24 + 38r + -20r2 Reorder the terms: 149 + 38r + -20r2 = -24 + 24 + -38r + 38r + 20r2 + -20r2 Combine like terms: -24 + 24 = 0 149 + 38r + -20r2 = 0 + -38r + 38r + 20r2 + -20r2 149 + 38r + -20r2 = -38r + 38r + 20r2 + -20r2 Combine like terms: -38r + 38r = 0 149 + 38r + -20r2 = 0 + 20r2 + -20r2 149 + 38r + -20r2 = 20r2 + -20r2 Combine like terms: 20r2 + -20r2 = 0 149 + 38r + -20r2 = 0 Begin completing the square. Divide all terms by -20 the coefficient of the squared term: Divide each side by '-20'. -7.45 + -1.9r + r2 = 0 Move the constant term to the right: Add '7.45' to each side of the equation. -7.45 + -1.9r + 7.45 + r2 = 0 + 7.45 Reorder the terms: -7.45 + 7.45 + -1.9r + r2 = 0 + 7.45 Combine like terms: -7.45 + 7.45 = 0.00 0.00 + -1.9r + r2 = 0 + 7.45 -1.9r + r2 = 0 + 7.45 Combine like terms: 0 + 7.45 = 7.45 -1.9r + r2 = 7.45 The r term is -1.9r. Take half its coefficient (-0.95). Square it (0.9025) and add it to both sides. Add '0.9025' to each side of the equation. -1.9r + 0.9025 + r2 = 7.45 + 0.9025 Reorder the terms: 0.9025 + -1.9r + r2 = 7.45 + 0.9025 Combine like terms: 7.45 + 0.9025 = 8.3525 0.9025 + -1.9r + r2 = 8.3525 Factor a perfect square on the left side: (r + -0.95)(r + -0.95) = 8.3525 Calculate the square root of the right side: 2.890069203 Break this problem into two subproblems by setting (r + -0.95) equal to 2.890069203 and -2.890069203.Subproblem 1
r + -0.95 = 2.890069203 Simplifying r + -0.95 = 2.890069203 Reorder the terms: -0.95 + r = 2.890069203 Solving -0.95 + r = 2.890069203 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.95' to each side of the equation. -0.95 + 0.95 + r = 2.890069203 + 0.95 Combine like terms: -0.95 + 0.95 = 0.00 0.00 + r = 2.890069203 + 0.95 r = 2.890069203 + 0.95 Combine like terms: 2.890069203 + 0.95 = 3.840069203 r = 3.840069203 Simplifying r = 3.840069203Subproblem 2
r + -0.95 = -2.890069203 Simplifying r + -0.95 = -2.890069203 Reorder the terms: -0.95 + r = -2.890069203 Solving -0.95 + r = -2.890069203 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.95' to each side of the equation. -0.95 + 0.95 + r = -2.890069203 + 0.95 Combine like terms: -0.95 + 0.95 = 0.00 0.00 + r = -2.890069203 + 0.95 r = -2.890069203 + 0.95 Combine like terms: -2.890069203 + 0.95 = -1.940069203 r = -1.940069203 Simplifying r = -1.940069203Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.840069203, -1.940069203}
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