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Simplifying 4x2 + -2x + 78 = 180 Reorder the terms: 78 + -2x + 4x2 = 180 Solving 78 + -2x + 4x2 = 180 Solving for variable 'x'. Reorder the terms: 78 + -180 + -2x + 4x2 = 180 + -180 Combine like terms: 78 + -180 = -102 -102 + -2x + 4x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -102 + -2x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-51 + -1x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-51 + -1x + 2x2)' equal to zero and attempt to solve: Simplifying -51 + -1x + 2x2 = 0 Solving -51 + -1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -25.5 + -0.5x + x2 = 0 Move the constant term to the right: Add '25.5' to each side of the equation. -25.5 + -0.5x + 25.5 + x2 = 0 + 25.5 Reorder the terms: -25.5 + 25.5 + -0.5x + x2 = 0 + 25.5 Combine like terms: -25.5 + 25.5 = 0.0 0.0 + -0.5x + x2 = 0 + 25.5 -0.5x + x2 = 0 + 25.5 Combine like terms: 0 + 25.5 = 25.5 -0.5x + x2 = 25.5 The x term is -0.5x. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5x + 0.0625 + x2 = 25.5 + 0.0625 Reorder the terms: 0.0625 + -0.5x + x2 = 25.5 + 0.0625 Combine like terms: 25.5 + 0.0625 = 25.5625 0.0625 + -0.5x + x2 = 25.5625 Factor a perfect square on the left side: (x + -0.25)(x + -0.25) = 25.5625 Calculate the square root of the right side: 5.055937104 Break this problem into two subproblems by setting (x + -0.25) equal to 5.055937104 and -5.055937104.Subproblem 1
x + -0.25 = 5.055937104 Simplifying x + -0.25 = 5.055937104 Reorder the terms: -0.25 + x = 5.055937104 Solving -0.25 + x = 5.055937104 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 5.055937104 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 5.055937104 + 0.25 x = 5.055937104 + 0.25 Combine like terms: 5.055937104 + 0.25 = 5.305937104 x = 5.305937104 Simplifying x = 5.305937104Subproblem 2
x + -0.25 = -5.055937104 Simplifying x + -0.25 = -5.055937104 Reorder the terms: -0.25 + x = -5.055937104 Solving -0.25 + x = -5.055937104 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -5.055937104 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -5.055937104 + 0.25 x = -5.055937104 + 0.25 Combine like terms: -5.055937104 + 0.25 = -4.805937104 x = -4.805937104 Simplifying x = -4.805937104Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.305937104, -4.805937104}Solution
x = {5.305937104, -4.805937104}
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