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Simplifying x2 + 13x4 + 42x3 = 0 Reorder the terms: x2 + 42x3 + 13x4 = 0 Solving x2 + 42x3 + 13x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(1 + 42x + 13x2) = 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 + 42x + 13x2)' equal to zero and attempt to solve: Simplifying 1 + 42x + 13x2 = 0 Solving 1 + 42x + 13x2 = 0 Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. 0.07692307692 + 3.230769231x + x2 = 0 Move the constant term to the right: Add '-0.07692307692' to each side of the equation. 0.07692307692 + 3.230769231x + -0.07692307692 + x2 = 0 + -0.07692307692 Reorder the terms: 0.07692307692 + -0.07692307692 + 3.230769231x + x2 = 0 + -0.07692307692 Combine like terms: 0.07692307692 + -0.07692307692 = 0.00000000000 0.00000000000 + 3.230769231x + x2 = 0 + -0.07692307692 3.230769231x + x2 = 0 + -0.07692307692 Combine like terms: 0 + -0.07692307692 = -0.07692307692 3.230769231x + x2 = -0.07692307692 The x term is 3.230769231x. Take half its coefficient (1.615384616). Square it (2.609467458) and add it to both sides. Add '2.609467458' to each side of the equation. 3.230769231x + 2.609467458 + x2 = -0.07692307692 + 2.609467458 Reorder the terms: 2.609467458 + 3.230769231x + x2 = -0.07692307692 + 2.609467458 Combine like terms: -0.07692307692 + 2.609467458 = 2.53254438108 2.609467458 + 3.230769231x + x2 = 2.53254438108 Factor a perfect square on the left side: (x + 1.615384616)(x + 1.615384616) = 2.53254438108 Calculate the square root of the right side: 1.59139699 Break this problem into two subproblems by setting (x + 1.615384616) equal to 1.59139699 and -1.59139699.Subproblem 1
x + 1.615384616 = 1.59139699 Simplifying x + 1.615384616 = 1.59139699 Reorder the terms: 1.615384616 + x = 1.59139699 Solving 1.615384616 + x = 1.59139699 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.615384616' to each side of the equation. 1.615384616 + -1.615384616 + x = 1.59139699 + -1.615384616 Combine like terms: 1.615384616 + -1.615384616 = 0.000000000 0.000000000 + x = 1.59139699 + -1.615384616 x = 1.59139699 + -1.615384616 Combine like terms: 1.59139699 + -1.615384616 = -0.023987626 x = -0.023987626 Simplifying x = -0.023987626Subproblem 2
x + 1.615384616 = -1.59139699 Simplifying x + 1.615384616 = -1.59139699 Reorder the terms: 1.615384616 + x = -1.59139699 Solving 1.615384616 + x = -1.59139699 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.615384616' to each side of the equation. 1.615384616 + -1.615384616 + x = -1.59139699 + -1.615384616 Combine like terms: 1.615384616 + -1.615384616 = 0.000000000 0.000000000 + x = -1.59139699 + -1.615384616 x = -1.59139699 + -1.615384616 Combine like terms: -1.59139699 + -1.615384616 = -3.206781606 x = -3.206781606 Simplifying x = -3.206781606Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.023987626, -3.206781606}Solution
x = {0, -0.023987626, -3.206781606}
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