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Simplifying 5(2c + 7) * 3c = 7(c + 5) Reorder the terms: 5(7 + 2c) * 3c = 7(c + 5) Reorder the terms for easier multiplication: 5 * 3c(7 + 2c) = 7(c + 5) Multiply 5 * 3 15c(7 + 2c) = 7(c + 5) (7 * 15c + 2c * 15c) = 7(c + 5) (105c + 30c2) = 7(c + 5) Reorder the terms: 105c + 30c2 = 7(5 + c) 105c + 30c2 = (5 * 7 + c * 7) 105c + 30c2 = (35 + 7c) Solving 105c + 30c2 = 35 + 7c Solving for variable 'c'. Reorder the terms: -35 + 105c + -7c + 30c2 = 35 + 7c + -35 + -7c Combine like terms: 105c + -7c = 98c -35 + 98c + 30c2 = 35 + 7c + -35 + -7c Reorder the terms: -35 + 98c + 30c2 = 35 + -35 + 7c + -7c Combine like terms: 35 + -35 = 0 -35 + 98c + 30c2 = 0 + 7c + -7c -35 + 98c + 30c2 = 7c + -7c Combine like terms: 7c + -7c = 0 -35 + 98c + 30c2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. -1.166666667 + 3.266666667c + c2 = 0.0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + 3.266666667c + 1.166666667 + c2 = 0.0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + 3.266666667c + c2 = 0.0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + 3.266666667c + c2 = 0.0 + 1.166666667 3.266666667c + c2 = 0.0 + 1.166666667 Combine like terms: 0.0 + 1.166666667 = 1.166666667 3.266666667c + c2 = 1.166666667 The c term is 3.266666667c. Take half its coefficient (1.633333334). Square it (2.667777780) and add it to both sides. Add '2.667777780' to each side of the equation. 3.266666667c + 2.667777780 + c2 = 1.166666667 + 2.667777780 Reorder the terms: 2.667777780 + 3.266666667c + c2 = 1.166666667 + 2.667777780 Combine like terms: 1.166666667 + 2.667777780 = 3.834444447 2.667777780 + 3.266666667c + c2 = 3.834444447 Factor a perfect square on the left side: (c + 1.633333334)(c + 1.633333334) = 3.834444447 Calculate the square root of the right side: 1.958173753 Break this problem into two subproblems by setting (c + 1.633333334) equal to 1.958173753 and -1.958173753.Subproblem 1
c + 1.633333334 = 1.958173753 Simplifying c + 1.633333334 = 1.958173753 Reorder the terms: 1.633333334 + c = 1.958173753 Solving 1.633333334 + c = 1.958173753 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.633333334' to each side of the equation. 1.633333334 + -1.633333334 + c = 1.958173753 + -1.633333334 Combine like terms: 1.633333334 + -1.633333334 = 0.000000000 0.000000000 + c = 1.958173753 + -1.633333334 c = 1.958173753 + -1.633333334 Combine like terms: 1.958173753 + -1.633333334 = 0.324840419 c = 0.324840419 Simplifying c = 0.324840419Subproblem 2
c + 1.633333334 = -1.958173753 Simplifying c + 1.633333334 = -1.958173753 Reorder the terms: 1.633333334 + c = -1.958173753 Solving 1.633333334 + c = -1.958173753 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.633333334' to each side of the equation. 1.633333334 + -1.633333334 + c = -1.958173753 + -1.633333334 Combine like terms: 1.633333334 + -1.633333334 = 0.000000000 0.000000000 + c = -1.958173753 + -1.633333334 c = -1.958173753 + -1.633333334 Combine like terms: -1.958173753 + -1.633333334 = -3.591507087 c = -3.591507087 Simplifying c = -3.591507087Solution
The solution to the problem is based on the solutions from the subproblems. c = {0.324840419, -3.591507087}
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