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Simplifying 6x3 + 21x2 + 5x = 0 Reorder the terms: 5x + 21x2 + 6x3 = 0 Solving 5x + 21x2 + 6x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x'. x(5 + 21x + 6x2) = 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 '(5 + 21x + 6x2)' equal to zero and attempt to solve: Simplifying 5 + 21x + 6x2 = 0 Solving 5 + 21x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.8333333333 + 3.5x + x2 = 0 Move the constant term to the right: Add '-0.8333333333' to each side of the equation. 0.8333333333 + 3.5x + -0.8333333333 + x2 = 0 + -0.8333333333 Reorder the terms: 0.8333333333 + -0.8333333333 + 3.5x + x2 = 0 + -0.8333333333 Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000 0.0000000000 + 3.5x + x2 = 0 + -0.8333333333 3.5x + x2 = 0 + -0.8333333333 Combine like terms: 0 + -0.8333333333 = -0.8333333333 3.5x + x2 = -0.8333333333 The x term is 3.5x. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5x + 3.0625 + x2 = -0.8333333333 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = -0.8333333333 + 3.0625 Combine like terms: -0.8333333333 + 3.0625 = 2.2291666667 3.0625 + 3.5x + x2 = 2.2291666667 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 2.2291666667 Calculate the square root of the right side: 1.493039406 Break this problem into two subproblems by setting (x + 1.75) equal to 1.493039406 and -1.493039406.Subproblem 1
x + 1.75 = 1.493039406 Simplifying x + 1.75 = 1.493039406 Reorder the terms: 1.75 + x = 1.493039406 Solving 1.75 + x = 1.493039406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 1.493039406 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 1.493039406 + -1.75 x = 1.493039406 + -1.75 Combine like terms: 1.493039406 + -1.75 = -0.256960594 x = -0.256960594 Simplifying x = -0.256960594Subproblem 2
x + 1.75 = -1.493039406 Simplifying x + 1.75 = -1.493039406 Reorder the terms: 1.75 + x = -1.493039406 Solving 1.75 + x = -1.493039406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -1.493039406 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -1.493039406 + -1.75 x = -1.493039406 + -1.75 Combine like terms: -1.493039406 + -1.75 = -3.243039406 x = -3.243039406 Simplifying x = -3.243039406Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.256960594, -3.243039406}Solution
x = {0, -0.256960594, -3.243039406}
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