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Simplifying 6x4 + 21x3 + x2 = 0 Reorder the terms: x2 + 21x3 + 6x4 = 0 Solving x2 + 21x3 + 6x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(1 + 21x + 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 '(1 + 21x + 6x2)' equal to zero and attempt to solve: Simplifying 1 + 21x + 6x2 = 0 Solving 1 + 21x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.1666666667 + 3.5x + x2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + 3.5x + -0.1666666667 + x2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + 3.5x + x2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + 3.5x + x2 = 0 + -0.1666666667 3.5x + x2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 3.5x + x2 = -0.1666666667 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 = -0.1666666667 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = -0.1666666667 + 3.0625 Combine like terms: -0.1666666667 + 3.0625 = 2.8958333333 3.0625 + 3.5x + x2 = 2.8958333333 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 2.8958333333 Calculate the square root of the right side: 1.701714821 Break this problem into two subproblems by setting (x + 1.75) equal to 1.701714821 and -1.701714821.Subproblem 1
x + 1.75 = 1.701714821 Simplifying x + 1.75 = 1.701714821 Reorder the terms: 1.75 + x = 1.701714821 Solving 1.75 + x = 1.701714821 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 = 1.701714821 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 1.701714821 + -1.75 x = 1.701714821 + -1.75 Combine like terms: 1.701714821 + -1.75 = -0.048285179 x = -0.048285179 Simplifying x = -0.048285179Subproblem 2
x + 1.75 = -1.701714821 Simplifying x + 1.75 = -1.701714821 Reorder the terms: 1.75 + x = -1.701714821 Solving 1.75 + x = -1.701714821 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 = -1.701714821 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -1.701714821 + -1.75 x = -1.701714821 + -1.75 Combine like terms: -1.701714821 + -1.75 = -3.451714821 x = -3.451714821 Simplifying x = -3.451714821Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.048285179, -3.451714821}Solution
x = {0, -0.048285179, -3.451714821}
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