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Simplifying 6x4 + 29x3 + 16x2 = 0 Reorder the terms: 16x2 + 29x3 + 6x4 = 0 Solving 16x2 + 29x3 + 6x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(16 + 29x + 6x2) = 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 '(16 + 29x + 6x2)' equal to zero and attempt to solve: Simplifying 16 + 29x + 6x2 = 0 Solving 16 + 29x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 2.666666667 + 4.833333333x + x2 = 0 Move the constant term to the right: Add '-2.666666667' to each side of the equation. 2.666666667 + 4.833333333x + -2.666666667 + x2 = 0 + -2.666666667 Reorder the terms: 2.666666667 + -2.666666667 + 4.833333333x + x2 = 0 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + 4.833333333x + x2 = 0 + -2.666666667 4.833333333x + x2 = 0 + -2.666666667 Combine like terms: 0 + -2.666666667 = -2.666666667 4.833333333x + x2 = -2.666666667 The x term is 4.833333333x. Take half its coefficient (2.416666667). Square it (5.840277779) and add it to both sides. Add '5.840277779' to each side of the equation. 4.833333333x + 5.840277779 + x2 = -2.666666667 + 5.840277779 Reorder the terms: 5.840277779 + 4.833333333x + x2 = -2.666666667 + 5.840277779 Combine like terms: -2.666666667 + 5.840277779 = 3.173611112 5.840277779 + 4.833333333x + x2 = 3.173611112 Factor a perfect square on the left side: (x + 2.416666667)(x + 2.416666667) = 3.173611112 Calculate the square root of the right side: 1.781463194 Break this problem into two subproblems by setting (x + 2.416666667) equal to 1.781463194 and -1.781463194.Subproblem 1
x + 2.416666667 = 1.781463194 Simplifying x + 2.416666667 = 1.781463194 Reorder the terms: 2.416666667 + x = 1.781463194 Solving 2.416666667 + x = 1.781463194 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + x = 1.781463194 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + x = 1.781463194 + -2.416666667 x = 1.781463194 + -2.416666667 Combine like terms: 1.781463194 + -2.416666667 = -0.635203473 x = -0.635203473 Simplifying x = -0.635203473Subproblem 2
x + 2.416666667 = -1.781463194 Simplifying x + 2.416666667 = -1.781463194 Reorder the terms: 2.416666667 + x = -1.781463194 Solving 2.416666667 + x = -1.781463194 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + x = -1.781463194 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + x = -1.781463194 + -2.416666667 x = -1.781463194 + -2.416666667 Combine like terms: -1.781463194 + -2.416666667 = -4.198129861 x = -4.198129861 Simplifying x = -4.198129861Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.635203473, -4.198129861}Solution
x = {0, -0.635203473, -4.198129861}
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