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Simplifying 54x2 + 66x + -135 = 0 Reorder the terms: -135 + 66x + 54x2 = 0 Solving -135 + 66x + 54x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3'. 3(-45 + 22x + 18x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-45 + 22x + 18x2)' equal to zero and attempt to solve: Simplifying -45 + 22x + 18x2 = 0 Solving -45 + 22x + 18x2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -2.5 + 1.222222222x + x2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 1.222222222x + 2.5 + x2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 1.222222222x + x2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 1.222222222x + x2 = 0 + 2.5 1.222222222x + x2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 1.222222222x + x2 = 2.5 The x term is 1.222222222x. Take half its coefficient (0.611111111). Square it (0.3734567900) and add it to both sides. Add '0.3734567900' to each side of the equation. 1.222222222x + 0.3734567900 + x2 = 2.5 + 0.3734567900 Reorder the terms: 0.3734567900 + 1.222222222x + x2 = 2.5 + 0.3734567900 Combine like terms: 2.5 + 0.3734567900 = 2.87345679 0.3734567900 + 1.222222222x + x2 = 2.87345679 Factor a perfect square on the left side: (x + 0.611111111)(x + 0.611111111) = 2.87345679 Calculate the square root of the right side: 1.695127367 Break this problem into two subproblems by setting (x + 0.611111111) equal to 1.695127367 and -1.695127367.Subproblem 1
x + 0.611111111 = 1.695127367 Simplifying x + 0.611111111 = 1.695127367 Reorder the terms: 0.611111111 + x = 1.695127367 Solving 0.611111111 + x = 1.695127367 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.611111111' to each side of the equation. 0.611111111 + -0.611111111 + x = 1.695127367 + -0.611111111 Combine like terms: 0.611111111 + -0.611111111 = 0.000000000 0.000000000 + x = 1.695127367 + -0.611111111 x = 1.695127367 + -0.611111111 Combine like terms: 1.695127367 + -0.611111111 = 1.084016256 x = 1.084016256 Simplifying x = 1.084016256Subproblem 2
x + 0.611111111 = -1.695127367 Simplifying x + 0.611111111 = -1.695127367 Reorder the terms: 0.611111111 + x = -1.695127367 Solving 0.611111111 + x = -1.695127367 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.611111111' to each side of the equation. 0.611111111 + -0.611111111 + x = -1.695127367 + -0.611111111 Combine like terms: 0.611111111 + -0.611111111 = 0.000000000 0.000000000 + x = -1.695127367 + -0.611111111 x = -1.695127367 + -0.611111111 Combine like terms: -1.695127367 + -0.611111111 = -2.306238478 x = -2.306238478 Simplifying x = -2.306238478Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.084016256, -2.306238478}Solution
x = {1.084016256, -2.306238478}
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