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Simplifying 0 = 5x4 + -10x3 + 2x2 Reorder the terms: 0 = 2x2 + -10x3 + 5x4 Solving 0 = 2x2 + -10x3 + 5x4 Solving for variable 'x'. Remove the zero: -2x2 + 10x3 + -5x4 = 2x2 + -10x3 + 5x4 + -2x2 + 10x3 + -5x4 Reorder the terms: -2x2 + 10x3 + -5x4 = 2x2 + -2x2 + -10x3 + 10x3 + 5x4 + -5x4 Combine like terms: 2x2 + -2x2 = 0 -2x2 + 10x3 + -5x4 = 0 + -10x3 + 10x3 + 5x4 + -5x4 -2x2 + 10x3 + -5x4 = -10x3 + 10x3 + 5x4 + -5x4 Combine like terms: -10x3 + 10x3 = 0 -2x2 + 10x3 + -5x4 = 0 + 5x4 + -5x4 -2x2 + 10x3 + -5x4 = 5x4 + -5x4 Combine like terms: 5x4 + -5x4 = 0 -2x2 + 10x3 + -5x4 = 0 Factor out the Greatest Common Factor (GCF), 'x2'. x2(-2 + 10x + -5x2) = 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 '(-2 + 10x + -5x2)' equal to zero and attempt to solve: Simplifying -2 + 10x + -5x2 = 0 Solving -2 + 10x + -5x2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 0.4 + -2x + x2 = 0 Move the constant term to the right: Add '-0.4' to each side of the equation. 0.4 + -2x + -0.4 + x2 = 0 + -0.4 Reorder the terms: 0.4 + -0.4 + -2x + x2 = 0 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + -2x + x2 = 0 + -0.4 -2x + x2 = 0 + -0.4 Combine like terms: 0 + -0.4 = -0.4 -2x + x2 = -0.4 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = -0.4 + 1 Reorder the terms: 1 + -2x + x2 = -0.4 + 1 Combine like terms: -0.4 + 1 = 0.6 1 + -2x + x2 = 0.6 Factor a perfect square on the left side: (x + -1)(x + -1) = 0.6 Calculate the square root of the right side: 0.774596669 Break this problem into two subproblems by setting (x + -1) equal to 0.774596669 and -0.774596669.Subproblem 1
x + -1 = 0.774596669 Simplifying x + -1 = 0.774596669 Reorder the terms: -1 + x = 0.774596669 Solving -1 + x = 0.774596669 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 0.774596669 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.774596669 + 1 x = 0.774596669 + 1 Combine like terms: 0.774596669 + 1 = 1.774596669 x = 1.774596669 Simplifying x = 1.774596669Subproblem 2
x + -1 = -0.774596669 Simplifying x + -1 = -0.774596669 Reorder the terms: -1 + x = -0.774596669 Solving -1 + x = -0.774596669 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -0.774596669 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.774596669 + 1 x = -0.774596669 + 1 Combine like terms: -0.774596669 + 1 = 0.225403331 x = 0.225403331 Simplifying x = 0.225403331Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.774596669, 0.225403331}Solution
x = {0, 1.774596669, 0.225403331}
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