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Simplifying 3x4 + 4x3 + -12x2 = 0 Reorder the terms: -12x2 + 4x3 + 3x4 = 0 Solving -12x2 + 4x3 + 3x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(-12 + 4x + 3x2) = 0Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve: Simplifying x2 = 0 Solving x2 = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x2 = 0 Take the square root of each side: x = {0}Subproblem 2
Set the factor '(-12 + 4x + 3x2)' equal to zero and attempt to solve: Simplifying -12 + 4x + 3x2 = 0 Solving -12 + 4x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -4 + 1.333333333x + x2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 1.333333333x + 4 + x2 = 0 + 4 Reorder the terms: -4 + 4 + 1.333333333x + x2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 1.333333333x + x2 = 0 + 4 1.333333333x + x2 = 0 + 4 Combine like terms: 0 + 4 = 4 1.333333333x + x2 = 4 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 = 4 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333x + x2 = 4 + 0.4444444442 Combine like terms: 4 + 0.4444444442 = 4.4444444442 0.4444444442 + 1.333333333x + x2 = 4.4444444442 Factor a perfect square on the left side: (x + 0.6666666665)(x + 0.6666666665) = 4.4444444442 Calculate the square root of the right side: 2.108185107 Break this problem into two subproblems by setting (x + 0.6666666665) equal to 2.108185107 and -2.108185107.Subproblem 1
x + 0.6666666665 = 2.108185107 Simplifying x + 0.6666666665 = 2.108185107 Reorder the terms: 0.6666666665 + x = 2.108185107 Solving 0.6666666665 + x = 2.108185107 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 = 2.108185107 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 2.108185107 + -0.6666666665 x = 2.108185107 + -0.6666666665 Combine like terms: 2.108185107 + -0.6666666665 = 1.4415184405 x = 1.4415184405 Simplifying x = 1.4415184405Subproblem 2
x + 0.6666666665 = -2.108185107 Simplifying x + 0.6666666665 = -2.108185107 Reorder the terms: 0.6666666665 + x = -2.108185107 Solving 0.6666666665 + x = -2.108185107 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 = -2.108185107 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -2.108185107 + -0.6666666665 x = -2.108185107 + -0.6666666665 Combine like terms: -2.108185107 + -0.6666666665 = -2.7748517735 x = -2.7748517735 Simplifying x = -2.7748517735Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.4415184405, -2.7748517735}Solution
x = {0, 1.4415184405, -2.7748517735}
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