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Simplifying 12x2 + 16x + -420 = 0 Reorder the terms: -420 + 16x + 12x2 = 0 Solving -420 + 16x + 12x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '4'. 4(-105 + 4x + 3x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-105 + 4x + 3x2)' equal to zero and attempt to solve: Simplifying -105 + 4x + 3x2 = 0 Solving -105 + 4x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -35 + 1.333333333x + x2 = 0 Move the constant term to the right: Add '35' to each side of the equation. -35 + 1.333333333x + 35 + x2 = 0 + 35 Reorder the terms: -35 + 35 + 1.333333333x + x2 = 0 + 35 Combine like terms: -35 + 35 = 0 0 + 1.333333333x + x2 = 0 + 35 1.333333333x + x2 = 0 + 35 Combine like terms: 0 + 35 = 35 1.333333333x + x2 = 35 The x term is 1.333333333x. Take half its coefficient (0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. 1.333333333x + 0.4444444442 + x2 = 35 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333x + x2 = 35 + 0.4444444442 Combine like terms: 35 + 0.4444444442 = 35.4444444442 0.4444444442 + 1.333333333x + x2 = 35.4444444442 Factor a perfect square on the left side: (x + 0.6666666665)(x + 0.6666666665) = 35.4444444442 Calculate the square root of the right side: 5.9535237 Break this problem into two subproblems by setting (x + 0.6666666665) equal to 5.9535237 and -5.9535237.Subproblem 1
x + 0.6666666665 = 5.9535237 Simplifying x + 0.6666666665 = 5.9535237 Reorder the terms: 0.6666666665 + x = 5.9535237 Solving 0.6666666665 + x = 5.9535237 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = 5.9535237 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 5.9535237 + -0.6666666665 x = 5.9535237 + -0.6666666665 Combine like terms: 5.9535237 + -0.6666666665 = 5.2868570335 x = 5.2868570335 Simplifying x = 5.2868570335Subproblem 2
x + 0.6666666665 = -5.9535237 Simplifying x + 0.6666666665 = -5.9535237 Reorder the terms: 0.6666666665 + x = -5.9535237 Solving 0.6666666665 + x = -5.9535237 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = -5.9535237 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -5.9535237 + -0.6666666665 x = -5.9535237 + -0.6666666665 Combine like terms: -5.9535237 + -0.6666666665 = -6.6201903665 x = -6.6201903665 Simplifying x = -6.6201903665Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.2868570335, -6.6201903665}Solution
x = {5.2868570335, -6.6201903665}
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