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Simplifying (7c + -7)(2c + -2) = 4c Reorder the terms: (-7 + 7c)(2c + -2) = 4c Reorder the terms: (-7 + 7c)(-2 + 2c) = 4c Multiply (-7 + 7c) * (-2 + 2c) (-7(-2 + 2c) + 7c * (-2 + 2c)) = 4c ((-2 * -7 + 2c * -7) + 7c * (-2 + 2c)) = 4c ((14 + -14c) + 7c * (-2 + 2c)) = 4c (14 + -14c + (-2 * 7c + 2c * 7c)) = 4c (14 + -14c + (-14c + 14c2)) = 4c Combine like terms: -14c + -14c = -28c (14 + -28c + 14c2) = 4c Solving 14 + -28c + 14c2 = 4c Solving for variable 'c'. Reorder the terms: 14 + -28c + -4c + 14c2 = 4c + -4c Combine like terms: -28c + -4c = -32c 14 + -32c + 14c2 = 4c + -4c Combine like terms: 4c + -4c = 0 14 + -32c + 14c2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -16c + 7c2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -16c + 7c2)' equal to zero and attempt to solve: Simplifying 7 + -16c + 7c2 = 0 Solving 7 + -16c + 7c2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 1 + -2.285714286c + c2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.285714286c + -1 + c2 = 0 + -1 Reorder the terms: 1 + -1 + -2.285714286c + c2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.285714286c + c2 = 0 + -1 -2.285714286c + c2 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.285714286c + c2 = -1 The c term is -2.285714286c. Take half its coefficient (-1.142857143). Square it (1.306122449) and add it to both sides. Add '1.306122449' to each side of the equation. -2.285714286c + 1.306122449 + c2 = -1 + 1.306122449 Reorder the terms: 1.306122449 + -2.285714286c + c2 = -1 + 1.306122449 Combine like terms: -1 + 1.306122449 = 0.306122449 1.306122449 + -2.285714286c + c2 = 0.306122449 Factor a perfect square on the left side: (c + -1.142857143)(c + -1.142857143) = 0.306122449 Calculate the square root of the right side: 0.553283335 Break this problem into two subproblems by setting (c + -1.142857143) equal to 0.553283335 and -0.553283335.Subproblem 1
c + -1.142857143 = 0.553283335 Simplifying c + -1.142857143 = 0.553283335 Reorder the terms: -1.142857143 + c = 0.553283335 Solving -1.142857143 + c = 0.553283335 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + c = 0.553283335 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + c = 0.553283335 + 1.142857143 c = 0.553283335 + 1.142857143 Combine like terms: 0.553283335 + 1.142857143 = 1.696140478 c = 1.696140478 Simplifying c = 1.696140478Subproblem 2
c + -1.142857143 = -0.553283335 Simplifying c + -1.142857143 = -0.553283335 Reorder the terms: -1.142857143 + c = -0.553283335 Solving -1.142857143 + c = -0.553283335 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + c = -0.553283335 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + c = -0.553283335 + 1.142857143 c = -0.553283335 + 1.142857143 Combine like terms: -0.553283335 + 1.142857143 = 0.589573808 c = 0.589573808 Simplifying c = 0.589573808Solution
The solution to the problem is based on the solutions from the subproblems. c = {1.696140478, 0.589573808}Solution
c = {1.696140478, 0.589573808}
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