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Simplifying 4x4 + -5x3 + -1x2 = 0 Reorder the terms: -1x2 + -5x3 + 4x4 = 0 Solving -1x2 + -5x3 + 4x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(-1 + -5x + 4x2) = 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 '(-1 + -5x + 4x2)' equal to zero and attempt to solve: Simplifying -1 + -5x + 4x2 = 0 Solving -1 + -5x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + -1.25x + x2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1.25x + 0.25 + x2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1.25x + x2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1.25x + x2 = 0 + 0.25 -1.25x + x2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1.25x + x2 = 0.25 The x term is -1.25x. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25x + 0.390625 + x2 = 0.25 + 0.390625 Reorder the terms: 0.390625 + -1.25x + x2 = 0.25 + 0.390625 Combine like terms: 0.25 + 0.390625 = 0.640625 0.390625 + -1.25x + x2 = 0.640625 Factor a perfect square on the left side: (x + -0.625)(x + -0.625) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (x + -0.625) equal to 0.80039053 and -0.80039053.Subproblem 1
x + -0.625 = 0.80039053 Simplifying x + -0.625 = 0.80039053 Reorder the terms: -0.625 + x = 0.80039053 Solving -0.625 + x = 0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = 0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = 0.80039053 + 0.625 x = 0.80039053 + 0.625 Combine like terms: 0.80039053 + 0.625 = 1.42539053 x = 1.42539053 Simplifying x = 1.42539053Subproblem 2
x + -0.625 = -0.80039053 Simplifying x + -0.625 = -0.80039053 Reorder the terms: -0.625 + x = -0.80039053 Solving -0.625 + x = -0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = -0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = -0.80039053 + 0.625 x = -0.80039053 + 0.625 Combine like terms: -0.80039053 + 0.625 = -0.17539053 x = -0.17539053 Simplifying x = -0.17539053Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.42539053, -0.17539053}Solution
x = {0, 1.42539053, -0.17539053}
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