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Simplifying 0 = 4x2 + 78x + -180 Reorder the terms: 0 = -180 + 78x + 4x2 Solving 0 = -180 + 78x + 4x2 Solving for variable 'x'. Combine like terms: 0 + 180 = 180 180 + -78x + -4x2 = -180 + 78x + 4x2 + 180 + -78x + -4x2 Reorder the terms: 180 + -78x + -4x2 = -180 + 180 + 78x + -78x + 4x2 + -4x2 Combine like terms: -180 + 180 = 0 180 + -78x + -4x2 = 0 + 78x + -78x + 4x2 + -4x2 180 + -78x + -4x2 = 78x + -78x + 4x2 + -4x2 Combine like terms: 78x + -78x = 0 180 + -78x + -4x2 = 0 + 4x2 + -4x2 180 + -78x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 180 + -78x + -4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(90 + -39x + -2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(90 + -39x + -2x2)' equal to zero and attempt to solve: Simplifying 90 + -39x + -2x2 = 0 Solving 90 + -39x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -45 + 19.5x + x2 = 0 Move the constant term to the right: Add '45' to each side of the equation. -45 + 19.5x + 45 + x2 = 0 + 45 Reorder the terms: -45 + 45 + 19.5x + x2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + 19.5x + x2 = 0 + 45 19.5x + x2 = 0 + 45 Combine like terms: 0 + 45 = 45 19.5x + x2 = 45 The x term is 19.5x. Take half its coefficient (9.75). Square it (95.0625) and add it to both sides. Add '95.0625' to each side of the equation. 19.5x + 95.0625 + x2 = 45 + 95.0625 Reorder the terms: 95.0625 + 19.5x + x2 = 45 + 95.0625 Combine like terms: 45 + 95.0625 = 140.0625 95.0625 + 19.5x + x2 = 140.0625 Factor a perfect square on the left side: (x + 9.75)(x + 9.75) = 140.0625 Calculate the square root of the right side: 11.834800379 Break this problem into two subproblems by setting (x + 9.75) equal to 11.834800379 and -11.834800379.Subproblem 1
x + 9.75 = 11.834800379 Simplifying x + 9.75 = 11.834800379 Reorder the terms: 9.75 + x = 11.834800379 Solving 9.75 + x = 11.834800379 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-9.75' to each side of the equation. 9.75 + -9.75 + x = 11.834800379 + -9.75 Combine like terms: 9.75 + -9.75 = 0.00 0.00 + x = 11.834800379 + -9.75 x = 11.834800379 + -9.75 Combine like terms: 11.834800379 + -9.75 = 2.084800379 x = 2.084800379 Simplifying x = 2.084800379Subproblem 2
x + 9.75 = -11.834800379 Simplifying x + 9.75 = -11.834800379 Reorder the terms: 9.75 + x = -11.834800379 Solving 9.75 + x = -11.834800379 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-9.75' to each side of the equation. 9.75 + -9.75 + x = -11.834800379 + -9.75 Combine like terms: 9.75 + -9.75 = 0.00 0.00 + x = -11.834800379 + -9.75 x = -11.834800379 + -9.75 Combine like terms: -11.834800379 + -9.75 = -21.584800379 x = -21.584800379 Simplifying x = -21.584800379Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.084800379, -21.584800379}Solution
x = {2.084800379, -21.584800379}
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