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Simplifying 32x4 + 16x3 + -80x2 = 0 Reorder the terms: -80x2 + 16x3 + 32x4 = 0 Solving -80x2 + 16x3 + 32x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '16x2'. 16x2(-5 + x + 2x2) = 0 Ignore the factor 16.Subproblem 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 '(-5 + x + 2x2)' equal to zero and attempt to solve: Simplifying -5 + x + 2x2 = 0 Solving -5 + x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.5 + 0.5x + x2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 0.5x + 2.5 + x2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 0.5x + x2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 0.5x + x2 = 0 + 2.5 0.5x + x2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 0.5x + x2 = 2.5 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5x + 0.25 + x2 = 2.5 + 0.25 Reorder the terms: 0.25 + 0.5x + x2 = 2.5 + 0.25 Combine like terms: 2.5 + 0.25 = 2.75 0.25 + 0.5x + x2 = 2.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (x + 0.5) equal to 1.658312395 and -1.658312395.Subproblem 1
x + 0.5 = 1.658312395 Simplifying x + 0.5 = 1.658312395 Reorder the terms: 0.5 + x = 1.658312395 Solving 0.5 + x = 1.658312395 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.658312395 + -0.5 x = 1.658312395 + -0.5 Combine like terms: 1.658312395 + -0.5 = 1.158312395 x = 1.158312395 Simplifying x = 1.158312395Subproblem 2
x + 0.5 = -1.658312395 Simplifying x + 0.5 = -1.658312395 Reorder the terms: 0.5 + x = -1.658312395 Solving 0.5 + x = -1.658312395 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.658312395 + -0.5 x = -1.658312395 + -0.5 Combine like terms: -1.658312395 + -0.5 = -2.158312395 x = -2.158312395 Simplifying x = -2.158312395Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.158312395, -2.158312395}Solution
x = {0, 1.158312395, -2.158312395}
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