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Simplifying 0 = 2x3 + 3x2 + -1680x Reorder the terms: 0 = -1680x + 3x2 + 2x3 Solving 0 = -1680x + 3x2 + 2x3 Solving for variable 'x'. Remove the zero: 1680x + -3x2 + -2x3 = -1680x + 3x2 + 2x3 + 1680x + -3x2 + -2x3 Reorder the terms: 1680x + -3x2 + -2x3 = -1680x + 1680x + 3x2 + -3x2 + 2x3 + -2x3 Combine like terms: -1680x + 1680x = 0 1680x + -3x2 + -2x3 = 0 + 3x2 + -3x2 + 2x3 + -2x3 1680x + -3x2 + -2x3 = 3x2 + -3x2 + 2x3 + -2x3 Combine like terms: 3x2 + -3x2 = 0 1680x + -3x2 + -2x3 = 0 + 2x3 + -2x3 1680x + -3x2 + -2x3 = 2x3 + -2x3 Combine like terms: 2x3 + -2x3 = 0 1680x + -3x2 + -2x3 = 0 Factor out the Greatest Common Factor (GCF), 'x'. x(1680 + -3x + -2x2) = 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 '(1680 + -3x + -2x2)' equal to zero and attempt to solve: Simplifying 1680 + -3x + -2x2 = 0 Solving 1680 + -3x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -840 + 1.5x + x2 = 0 Move the constant term to the right: Add '840' to each side of the equation. -840 + 1.5x + 840 + x2 = 0 + 840 Reorder the terms: -840 + 840 + 1.5x + x2 = 0 + 840 Combine like terms: -840 + 840 = 0 0 + 1.5x + x2 = 0 + 840 1.5x + x2 = 0 + 840 Combine like terms: 0 + 840 = 840 1.5x + x2 = 840 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 840 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 840 + 0.5625 Combine like terms: 840 + 0.5625 = 840.5625 0.5625 + 1.5x + x2 = 840.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 840.5625 Calculate the square root of the right side: 28.992455915 Break this problem into two subproblems by setting (x + 0.75) equal to 28.992455915 and -28.992455915.Subproblem 1
x + 0.75 = 28.992455915 Simplifying x + 0.75 = 28.992455915 Reorder the terms: 0.75 + x = 28.992455915 Solving 0.75 + x = 28.992455915 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 28.992455915 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 28.992455915 + -0.75 x = 28.992455915 + -0.75 Combine like terms: 28.992455915 + -0.75 = 28.242455915 x = 28.242455915 Simplifying x = 28.242455915Subproblem 2
x + 0.75 = -28.992455915 Simplifying x + 0.75 = -28.992455915 Reorder the terms: 0.75 + x = -28.992455915 Solving 0.75 + x = -28.992455915 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -28.992455915 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -28.992455915 + -0.75 x = -28.992455915 + -0.75 Combine like terms: -28.992455915 + -0.75 = -29.742455915 x = -29.742455915 Simplifying x = -29.742455915Solution
The solution to the problem is based on the solutions from the subproblems. x = {28.242455915, -29.742455915}Solution
x = {0, 28.242455915, -29.742455915}
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