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Simplifying (8c2 + -2c + -8) + -1(c2 + 12) = 0 Reorder the terms: (-8 + -2c + 8c2) + -1(c2 + 12) = 0 Remove parenthesis around (-8 + -2c + 8c2) -8 + -2c + 8c2 + -1(c2 + 12) = 0 Reorder the terms: -8 + -2c + 8c2 + -1(12 + c2) = 0 -8 + -2c + 8c2 + (12 * -1 + c2 * -1) = 0 -8 + -2c + 8c2 + (-12 + -1c2) = 0 Reorder the terms: -8 + -12 + -2c + 8c2 + -1c2 = 0 Combine like terms: -8 + -12 = -20 -20 + -2c + 8c2 + -1c2 = 0 Combine like terms: 8c2 + -1c2 = 7c2 -20 + -2c + 7c2 = 0 Solving -20 + -2c + 7c2 = 0 Solving for variable 'c'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.857142857 + -0.2857142857c + c2 = 0 Move the constant term to the right: Add '2.857142857' to each side of the equation. -2.857142857 + -0.2857142857c + 2.857142857 + c2 = 0 + 2.857142857 Reorder the terms: -2.857142857 + 2.857142857 + -0.2857142857c + c2 = 0 + 2.857142857 Combine like terms: -2.857142857 + 2.857142857 = 0.000000000 0.000000000 + -0.2857142857c + c2 = 0 + 2.857142857 -0.2857142857c + c2 = 0 + 2.857142857 Combine like terms: 0 + 2.857142857 = 2.857142857 -0.2857142857c + c2 = 2.857142857 The c term is -0.2857142857c. Take half its coefficient (-0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. -0.2857142857c + 0.02040816328 + c2 = 2.857142857 + 0.02040816328 Reorder the terms: 0.02040816328 + -0.2857142857c + c2 = 2.857142857 + 0.02040816328 Combine like terms: 2.857142857 + 0.02040816328 = 2.87755102028 0.02040816328 + -0.2857142857c + c2 = 2.87755102028 Factor a perfect square on the left side: (c + -0.1428571429)(c + -0.1428571429) = 2.87755102028 Calculate the square root of the right side: 1.696334584 Break this problem into two subproblems by setting (c + -0.1428571429) equal to 1.696334584 and -1.696334584.Subproblem 1
c + -0.1428571429 = 1.696334584 Simplifying c + -0.1428571429 = 1.696334584 Reorder the terms: -0.1428571429 + c = 1.696334584 Solving -0.1428571429 + c = 1.696334584 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + c = 1.696334584 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + c = 1.696334584 + 0.1428571429 c = 1.696334584 + 0.1428571429 Combine like terms: 1.696334584 + 0.1428571429 = 1.8391917269 c = 1.8391917269 Simplifying c = 1.8391917269Subproblem 2
c + -0.1428571429 = -1.696334584 Simplifying c + -0.1428571429 = -1.696334584 Reorder the terms: -0.1428571429 + c = -1.696334584 Solving -0.1428571429 + c = -1.696334584 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + c = -1.696334584 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + c = -1.696334584 + 0.1428571429 c = -1.696334584 + 0.1428571429 Combine like terms: -1.696334584 + 0.1428571429 = -1.5534774411 c = -1.5534774411 Simplifying c = -1.5534774411Solution
The solution to the problem is based on the solutions from the subproblems. c = {1.8391917269, -1.5534774411}
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