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Simplifying 18x4 + 2x3 + 2x2 = 0 Reorder the terms: 2x2 + 2x3 + 18x4 = 0 Solving 2x2 + 2x3 + 18x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2x2'. 2x2(1 + x + 9x2) = 0 Ignore the factor 2.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 '(1 + x + 9x2)' equal to zero and attempt to solve: Simplifying 1 + x + 9x2 = 0 Solving 1 + x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.1111111111 + 0.1111111111x + x2 = 0 Move the constant term to the right: Add '-0.1111111111' to each side of the equation. 0.1111111111 + 0.1111111111x + -0.1111111111 + x2 = 0 + -0.1111111111 Reorder the terms: 0.1111111111 + -0.1111111111 + 0.1111111111x + x2 = 0 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + 0.1111111111x + x2 = 0 + -0.1111111111 0.1111111111x + x2 = 0 + -0.1111111111 Combine like terms: 0 + -0.1111111111 = -0.1111111111 0.1111111111x + x2 = -0.1111111111 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. 0.1111111111x + 0.25 + x2 = -0.1111111111 + 0.25 Reorder the terms: 0.25 + 0.1111111111x + x2 = -0.1111111111 + 0.25 Combine like terms: -0.1111111111 + 0.25 = 0.1388888889 0.25 + 0.1111111111x + x2 = 0.1388888889 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.1388888889 Calculate the square root of the right side: 0.372677996 Break this problem into two subproblems by setting (x + 0.5) equal to 0.372677996 and -0.372677996.Subproblem 1
x + 0.5 = 0.372677996 Simplifying x + 0.5 = 0.372677996 Reorder the terms: 0.5 + x = 0.372677996 Solving 0.5 + x = 0.372677996 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 = 0.372677996 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.372677996 + -0.5 x = 0.372677996 + -0.5 Combine like terms: 0.372677996 + -0.5 = -0.127322004 x = -0.127322004 Simplifying x = -0.127322004Subproblem 2
x + 0.5 = -0.372677996 Simplifying x + 0.5 = -0.372677996 Reorder the terms: 0.5 + x = -0.372677996 Solving 0.5 + x = -0.372677996 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 = -0.372677996 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.372677996 + -0.5 x = -0.372677996 + -0.5 Combine like terms: -0.372677996 + -0.5 = -0.872677996 x = -0.872677996 Simplifying x = -0.872677996Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.127322004, -0.872677996}Solution
x = {0, -0.127322004, -0.872677996}
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