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Simplifying 135 = 16x2 + 110x + 5 Reorder the terms: 135 = 5 + 110x + 16x2 Solving 135 = 5 + 110x + 16x2 Solving for variable 'x'. Combine like terms: 135 + -5 = 130 130 + -110x + -16x2 = 5 + 110x + 16x2 + -5 + -110x + -16x2 Reorder the terms: 130 + -110x + -16x2 = 5 + -5 + 110x + -110x + 16x2 + -16x2 Combine like terms: 5 + -5 = 0 130 + -110x + -16x2 = 0 + 110x + -110x + 16x2 + -16x2 130 + -110x + -16x2 = 110x + -110x + 16x2 + -16x2 Combine like terms: 110x + -110x = 0 130 + -110x + -16x2 = 0 + 16x2 + -16x2 130 + -110x + -16x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 130 + -110x + -16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(65 + -55x + -8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(65 + -55x + -8x2)' equal to zero and attempt to solve: Simplifying 65 + -55x + -8x2 = 0 Solving 65 + -55x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -8.125 + 6.875x + x2 = 0 Move the constant term to the right: Add '8.125' to each side of the equation. -8.125 + 6.875x + 8.125 + x2 = 0 + 8.125 Reorder the terms: -8.125 + 8.125 + 6.875x + x2 = 0 + 8.125 Combine like terms: -8.125 + 8.125 = 0.000 0.000 + 6.875x + x2 = 0 + 8.125 6.875x + x2 = 0 + 8.125 Combine like terms: 0 + 8.125 = 8.125 6.875x + x2 = 8.125 The x term is 6.875x. Take half its coefficient (3.4375). Square it (11.81640625) and add it to both sides. Add '11.81640625' to each side of the equation. 6.875x + 11.81640625 + x2 = 8.125 + 11.81640625 Reorder the terms: 11.81640625 + 6.875x + x2 = 8.125 + 11.81640625 Combine like terms: 8.125 + 11.81640625 = 19.94140625 11.81640625 + 6.875x + x2 = 19.94140625 Factor a perfect square on the left side: (x + 3.4375)(x + 3.4375) = 19.94140625 Calculate the square root of the right side: 4.465580169 Break this problem into two subproblems by setting (x + 3.4375) equal to 4.465580169 and -4.465580169.Subproblem 1
x + 3.4375 = 4.465580169 Simplifying x + 3.4375 = 4.465580169 Reorder the terms: 3.4375 + x = 4.465580169 Solving 3.4375 + x = 4.465580169 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.4375' to each side of the equation. 3.4375 + -3.4375 + x = 4.465580169 + -3.4375 Combine like terms: 3.4375 + -3.4375 = 0.0000 0.0000 + x = 4.465580169 + -3.4375 x = 4.465580169 + -3.4375 Combine like terms: 4.465580169 + -3.4375 = 1.028080169 x = 1.028080169 Simplifying x = 1.028080169Subproblem 2
x + 3.4375 = -4.465580169 Simplifying x + 3.4375 = -4.465580169 Reorder the terms: 3.4375 + x = -4.465580169 Solving 3.4375 + x = -4.465580169 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.4375' to each side of the equation. 3.4375 + -3.4375 + x = -4.465580169 + -3.4375 Combine like terms: 3.4375 + -3.4375 = 0.0000 0.0000 + x = -4.465580169 + -3.4375 x = -4.465580169 + -3.4375 Combine like terms: -4.465580169 + -3.4375 = -7.903080169 x = -7.903080169 Simplifying x = -7.903080169Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.028080169, -7.903080169}Solution
x = {1.028080169, -7.903080169}
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