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Simplifying 7(8 + -1r) * 3(2 + 2r) = 9 Reorder the terms for easier multiplication: 7 * 3(8 + -1r)(2 + 2r) = 9 Multiply 7 * 3 21(8 + -1r)(2 + 2r) = 9 Multiply (8 + -1r) * (2 + 2r) 21(8(2 + 2r) + -1r * (2 + 2r)) = 9 21((2 * 8 + 2r * 8) + -1r * (2 + 2r)) = 9 21((16 + 16r) + -1r * (2 + 2r)) = 9 21(16 + 16r + (2 * -1r + 2r * -1r)) = 9 21(16 + 16r + (-2r + -2r2)) = 9 Combine like terms: 16r + -2r = 14r 21(16 + 14r + -2r2) = 9 (16 * 21 + 14r * 21 + -2r2 * 21) = 9 (336 + 294r + -42r2) = 9 Solving 336 + 294r + -42r2 = 9 Solving for variable 'r'. Reorder the terms: 336 + -9 + 294r + -42r2 = 9 + -9 Combine like terms: 336 + -9 = 327 327 + 294r + -42r2 = 9 + -9 Combine like terms: 9 + -9 = 0 327 + 294r + -42r2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(109 + 98r + -14r2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(109 + 98r + -14r2)' equal to zero and attempt to solve: Simplifying 109 + 98r + -14r2 = 0 Solving 109 + 98r + -14r2 = 0 Begin completing the square. Divide all terms by -14 the coefficient of the squared term: Divide each side by '-14'. -7.785714286 + -7r + r2 = 0 Move the constant term to the right: Add '7.785714286' to each side of the equation. -7.785714286 + -7r + 7.785714286 + r2 = 0 + 7.785714286 Reorder the terms: -7.785714286 + 7.785714286 + -7r + r2 = 0 + 7.785714286 Combine like terms: -7.785714286 + 7.785714286 = 0.000000000 0.000000000 + -7r + r2 = 0 + 7.785714286 -7r + r2 = 0 + 7.785714286 Combine like terms: 0 + 7.785714286 = 7.785714286 -7r + r2 = 7.785714286 The r term is -7r. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7r + 12.25 + r2 = 7.785714286 + 12.25 Reorder the terms: 12.25 + -7r + r2 = 7.785714286 + 12.25 Combine like terms: 7.785714286 + 12.25 = 20.035714286 12.25 + -7r + r2 = 20.035714286 Factor a perfect square on the left side: (r + -3.5)(r + -3.5) = 20.035714286 Calculate the square root of the right side: 4.476127153 Break this problem into two subproblems by setting (r + -3.5) equal to 4.476127153 and -4.476127153.Subproblem 1
r + -3.5 = 4.476127153 Simplifying r + -3.5 = 4.476127153 Reorder the terms: -3.5 + r = 4.476127153 Solving -3.5 + r = 4.476127153 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + r = 4.476127153 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + r = 4.476127153 + 3.5 r = 4.476127153 + 3.5 Combine like terms: 4.476127153 + 3.5 = 7.976127153 r = 7.976127153 Simplifying r = 7.976127153Subproblem 2
r + -3.5 = -4.476127153 Simplifying r + -3.5 = -4.476127153 Reorder the terms: -3.5 + r = -4.476127153 Solving -3.5 + r = -4.476127153 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + r = -4.476127153 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + r = -4.476127153 + 3.5 r = -4.476127153 + 3.5 Combine like terms: -4.476127153 + 3.5 = -0.976127153 r = -0.976127153 Simplifying r = -0.976127153Solution
The solution to the problem is based on the solutions from the subproblems. r = {7.976127153, -0.976127153}Solution
r = {7.976127153, -0.976127153}
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