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Simplifying (6x3 + -5x2 + 5x) + (9x3 + 7x2 + -12x) = 0 Reorder the terms: (5x + -5x2 + 6x3) + (9x3 + 7x2 + -12x) = 0 Remove parenthesis around (5x + -5x2 + 6x3) 5x + -5x2 + 6x3 + (9x3 + 7x2 + -12x) = 0 Reorder the terms: 5x + -5x2 + 6x3 + (-12x + 7x2 + 9x3) = 0 Remove parenthesis around (-12x + 7x2 + 9x3) 5x + -5x2 + 6x3 + -12x + 7x2 + 9x3 = 0 Reorder the terms: 5x + -12x + -5x2 + 7x2 + 6x3 + 9x3 = 0 Combine like terms: 5x + -12x = -7x -7x + -5x2 + 7x2 + 6x3 + 9x3 = 0 Combine like terms: -5x2 + 7x2 = 2x2 -7x + 2x2 + 6x3 + 9x3 = 0 Combine like terms: 6x3 + 9x3 = 15x3 -7x + 2x2 + 15x3 = 0 Solving -7x + 2x2 + 15x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x'. x(-7 + 2x + 15x2) = 0Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(-7 + 2x + 15x2)' equal to zero and attempt to solve: Simplifying -7 + 2x + 15x2 = 0 Solving -7 + 2x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.4666666667 + 0.1333333333x + x2 = 0 Move the constant term to the right: Add '0.4666666667' to each side of the equation. -0.4666666667 + 0.1333333333x + 0.4666666667 + x2 = 0 + 0.4666666667 Reorder the terms: -0.4666666667 + 0.4666666667 + 0.1333333333x + x2 = 0 + 0.4666666667 Combine like terms: -0.4666666667 + 0.4666666667 = 0.0000000000 0.0000000000 + 0.1333333333x + x2 = 0 + 0.4666666667 0.1333333333x + x2 = 0 + 0.4666666667 Combine like terms: 0 + 0.4666666667 = 0.4666666667 0.1333333333x + x2 = 0.4666666667 The x term is 0.1333333333x. Take half its coefficient (0.06666666665). Square it (0.004444444442) and add it to both sides. Add '0.004444444442' to each side of the equation. 0.1333333333x + 0.004444444442 + x2 = 0.4666666667 + 0.004444444442 Reorder the terms: 0.004444444442 + 0.1333333333x + x2 = 0.4666666667 + 0.004444444442 Combine like terms: 0.4666666667 + 0.004444444442 = 0.471111111142 0.004444444442 + 0.1333333333x + x2 = 0.471111111142 Factor a perfect square on the left side: (x + 0.06666666665)(x + 0.06666666665) = 0.471111111142 Calculate the square root of the right side: 0.686375343 Break this problem into two subproblems by setting (x + 0.06666666665) equal to 0.686375343 and -0.686375343.Subproblem 1
x + 0.06666666665 = 0.686375343 Simplifying x + 0.06666666665 = 0.686375343 Reorder the terms: 0.06666666665 + x = 0.686375343 Solving 0.06666666665 + x = 0.686375343 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.06666666665' to each side of the equation. 0.06666666665 + -0.06666666665 + x = 0.686375343 + -0.06666666665 Combine like terms: 0.06666666665 + -0.06666666665 = 0.00000000000 0.00000000000 + x = 0.686375343 + -0.06666666665 x = 0.686375343 + -0.06666666665 Combine like terms: 0.686375343 + -0.06666666665 = 0.61970867635 x = 0.61970867635 Simplifying x = 0.61970867635Subproblem 2
x + 0.06666666665 = -0.686375343 Simplifying x + 0.06666666665 = -0.686375343 Reorder the terms: 0.06666666665 + x = -0.686375343 Solving 0.06666666665 + x = -0.686375343 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.06666666665' to each side of the equation. 0.06666666665 + -0.06666666665 + x = -0.686375343 + -0.06666666665 Combine like terms: 0.06666666665 + -0.06666666665 = 0.00000000000 0.00000000000 + x = -0.686375343 + -0.06666666665 x = -0.686375343 + -0.06666666665 Combine like terms: -0.686375343 + -0.06666666665 = -0.75304200965 x = -0.75304200965 Simplifying x = -0.75304200965Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.61970867635, -0.75304200965}Solution
x = {0, 0.61970867635, -0.75304200965}
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