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Simplifying 0 = (6x + 7x)(x2 + -3x + 1) Combine like terms: 6x + 7x = 13x 0 = (13x)(x2 + -3x + 1) Remove parenthesis around (13x) 0 = 13x(x2 + -3x + 1) Reorder the terms: 0 = 13x(1 + -3x + x2) 0 = (1 * 13x + -3x * 13x + x2 * 13x) 0 = (13x + -39x2 + 13x3) Solving 0 = 13x + -39x2 + 13x3 Solving for variable 'x'. Remove the zero: -13x + 39x2 + -13x3 = 13x + -39x2 + 13x3 + -13x + 39x2 + -13x3 Reorder the terms: -13x + 39x2 + -13x3 = 13x + -13x + -39x2 + 39x2 + 13x3 + -13x3 Combine like terms: 13x + -13x = 0 -13x + 39x2 + -13x3 = 0 + -39x2 + 39x2 + 13x3 + -13x3 -13x + 39x2 + -13x3 = -39x2 + 39x2 + 13x3 + -13x3 Combine like terms: -39x2 + 39x2 = 0 -13x + 39x2 + -13x3 = 0 + 13x3 + -13x3 -13x + 39x2 + -13x3 = 13x3 + -13x3 Combine like terms: 13x3 + -13x3 = 0 -13x + 39x2 + -13x3 = 0 Factor out the Greatest Common Factor (GCF), '13x'. 13x(-1 + 3x + -1x2) = 0 Ignore the factor 13.Subproblem 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 '(-1 + 3x + -1x2)' equal to zero and attempt to solve: Simplifying -1 + 3x + -1x2 = 0 Solving -1 + 3x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1 + -3x + x2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -3x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + -3x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -3x + x2 = 0 + -1 -3x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 -3x + x2 = -1 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = -1 + 2.25 Reorder the terms: 2.25 + -3x + x2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + -3x + x2 = 1.25 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (x + -1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
x + -1.5 = 1.118033989 Simplifying x + -1.5 = 1.118033989 Reorder the terms: -1.5 + x = 1.118033989 Solving -1.5 + x = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 1.118033989 + 1.5 x = 1.118033989 + 1.5 Combine like terms: 1.118033989 + 1.5 = 2.618033989 x = 2.618033989 Simplifying x = 2.618033989Subproblem 2
x + -1.5 = -1.118033989 Simplifying x + -1.5 = -1.118033989 Reorder the terms: -1.5 + x = -1.118033989 Solving -1.5 + x = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -1.118033989 + 1.5 x = -1.118033989 + 1.5 Combine like terms: -1.118033989 + 1.5 = 0.381966011 x = 0.381966011 Simplifying x = 0.381966011Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.618033989, 0.381966011}Solution
x = {0, 2.618033989, 0.381966011}
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