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Simplifying (c + -6)(c + -3) = 23 Reorder the terms: (-6 + c)(c + -3) = 23 Reorder the terms: (-6 + c)(-3 + c) = 23 Multiply (-6 + c) * (-3 + c) (-6(-3 + c) + c(-3 + c)) = 23 ((-3 * -6 + c * -6) + c(-3 + c)) = 23 ((18 + -6c) + c(-3 + c)) = 23 (18 + -6c + (-3 * c + c * c)) = 23 (18 + -6c + (-3c + c2)) = 23 Combine like terms: -6c + -3c = -9c (18 + -9c + c2) = 23 Solving 18 + -9c + c2 = 23 Solving for variable 'c'. Reorder the terms: 18 + -23 + -9c + c2 = 23 + -23 Combine like terms: 18 + -23 = -5 -5 + -9c + c2 = 23 + -23 Combine like terms: 23 + -23 = 0 -5 + -9c + c2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + -9c + 5 + c2 = 0 + 5 Reorder the terms: -5 + 5 + -9c + c2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -9c + c2 = 0 + 5 -9c + c2 = 0 + 5 Combine like terms: 0 + 5 = 5 -9c + c2 = 5 The c term is -9c. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9c + 20.25 + c2 = 5 + 20.25 Reorder the terms: 20.25 + -9c + c2 = 5 + 20.25 Combine like terms: 5 + 20.25 = 25.25 20.25 + -9c + c2 = 25.25 Factor a perfect square on the left side: (c + -4.5)(c + -4.5) = 25.25 Calculate the square root of the right side: 5.024937811 Break this problem into two subproblems by setting (c + -4.5) equal to 5.024937811 and -5.024937811.Subproblem 1
c + -4.5 = 5.024937811 Simplifying c + -4.5 = 5.024937811 Reorder the terms: -4.5 + c = 5.024937811 Solving -4.5 + c = 5.024937811 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + c = 5.024937811 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + c = 5.024937811 + 4.5 c = 5.024937811 + 4.5 Combine like terms: 5.024937811 + 4.5 = 9.524937811 c = 9.524937811 Simplifying c = 9.524937811Subproblem 2
c + -4.5 = -5.024937811 Simplifying c + -4.5 = -5.024937811 Reorder the terms: -4.5 + c = -5.024937811 Solving -4.5 + c = -5.024937811 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + c = -5.024937811 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + c = -5.024937811 + 4.5 c = -5.024937811 + 4.5 Combine like terms: -5.024937811 + 4.5 = -0.524937811 c = -0.524937811 Simplifying c = -0.524937811Solution
The solution to the problem is based on the solutions from the subproblems. c = {9.524937811, -0.524937811}
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