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Simplifying (2c + -1)(c + -5) + (c + -4)(c + -3) = 0 Reorder the terms: (-1 + 2c)(c + -5) + (c + -4)(c + -3) = 0 Reorder the terms: (-1 + 2c)(-5 + c) + (c + -4)(c + -3) = 0 Multiply (-1 + 2c) * (-5 + c) (-1(-5 + c) + 2c * (-5 + c)) + (c + -4)(c + -3) = 0 ((-5 * -1 + c * -1) + 2c * (-5 + c)) + (c + -4)(c + -3) = 0 ((5 + -1c) + 2c * (-5 + c)) + (c + -4)(c + -3) = 0 (5 + -1c + (-5 * 2c + c * 2c)) + (c + -4)(c + -3) = 0 (5 + -1c + (-10c + 2c2)) + (c + -4)(c + -3) = 0 Combine like terms: -1c + -10c = -11c (5 + -11c + 2c2) + (c + -4)(c + -3) = 0 Reorder the terms: 5 + -11c + 2c2 + (-4 + c)(c + -3) = 0 Reorder the terms: 5 + -11c + 2c2 + (-4 + c)(-3 + c) = 0 Multiply (-4 + c) * (-3 + c) 5 + -11c + 2c2 + (-4(-3 + c) + c(-3 + c)) = 0 5 + -11c + 2c2 + ((-3 * -4 + c * -4) + c(-3 + c)) = 0 5 + -11c + 2c2 + ((12 + -4c) + c(-3 + c)) = 0 5 + -11c + 2c2 + (12 + -4c + (-3 * c + c * c)) = 0 5 + -11c + 2c2 + (12 + -4c + (-3c + c2)) = 0 Combine like terms: -4c + -3c = -7c 5 + -11c + 2c2 + (12 + -7c + c2) = 0 Reorder the terms: 5 + 12 + -11c + -7c + 2c2 + c2 = 0 Combine like terms: 5 + 12 = 17 17 + -11c + -7c + 2c2 + c2 = 0 Combine like terms: -11c + -7c = -18c 17 + -18c + 2c2 + c2 = 0 Combine like terms: 2c2 + c2 = 3c2 17 + -18c + 3c2 = 0 Solving 17 + -18c + 3c2 = 0 Solving for variable 'c'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 5.666666667 + -6c + c2 = 0 Move the constant term to the right: Add '-5.666666667' to each side of the equation. 5.666666667 + -6c + -5.666666667 + c2 = 0 + -5.666666667 Reorder the terms: 5.666666667 + -5.666666667 + -6c + c2 = 0 + -5.666666667 Combine like terms: 5.666666667 + -5.666666667 = 0.000000000 0.000000000 + -6c + c2 = 0 + -5.666666667 -6c + c2 = 0 + -5.666666667 Combine like terms: 0 + -5.666666667 = -5.666666667 -6c + c2 = -5.666666667 The c term is -6c. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6c + 9 + c2 = -5.666666667 + 9 Reorder the terms: 9 + -6c + c2 = -5.666666667 + 9 Combine like terms: -5.666666667 + 9 = 3.333333333 9 + -6c + c2 = 3.333333333 Factor a perfect square on the left side: (c + -3)(c + -3) = 3.333333333 Calculate the square root of the right side: 1.825741858 Break this problem into two subproblems by setting (c + -3) equal to 1.825741858 and -1.825741858.Subproblem 1
c + -3 = 1.825741858 Simplifying c + -3 = 1.825741858 Reorder the terms: -3 + c = 1.825741858 Solving -3 + c = 1.825741858 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + c = 1.825741858 + 3 Combine like terms: -3 + 3 = 0 0 + c = 1.825741858 + 3 c = 1.825741858 + 3 Combine like terms: 1.825741858 + 3 = 4.825741858 c = 4.825741858 Simplifying c = 4.825741858Subproblem 2
c + -3 = -1.825741858 Simplifying c + -3 = -1.825741858 Reorder the terms: -3 + c = -1.825741858 Solving -3 + c = -1.825741858 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + c = -1.825741858 + 3 Combine like terms: -3 + 3 = 0 0 + c = -1.825741858 + 3 c = -1.825741858 + 3 Combine like terms: -1.825741858 + 3 = 1.174258142 c = 1.174258142 Simplifying c = 1.174258142Solution
The solution to the problem is based on the solutions from the subproblems. c = {4.825741858, 1.174258142}
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