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Simplifying 4x3 + x2 + 0.65x = 0 Reorder the terms: 0.65x + x2 + 4x3 = 0 Solving 0.65x + x2 + 4x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x'. x(0.65 + x + 4x2) = 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 '(0.65 + x + 4x2)' equal to zero and attempt to solve: Simplifying 0.65 + x + 4x2 = 0 Solving 0.65 + x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.1625 + 0.25x + x2 = 0 Move the constant term to the right: Add '-0.1625' to each side of the equation. 0.1625 + 0.25x + -0.1625 + x2 = 0 + -0.1625 Reorder the terms: 0.1625 + -0.1625 + 0.25x + x2 = 0 + -0.1625 Combine like terms: 0.1625 + -0.1625 = 0.0000 0.0000 + 0.25x + x2 = 0 + -0.1625 0.25x + x2 = 0 + -0.1625 Combine like terms: 0 + -0.1625 = -0.1625 0.25x + x2 = -0.1625 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. 0.25x + 0.25 + x2 = -0.1625 + 0.25 Reorder the terms: 0.25 + 0.25x + x2 = -0.1625 + 0.25 Combine like terms: -0.1625 + 0.25 = 0.0875 0.25 + 0.25x + x2 = 0.0875 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.0875 Calculate the square root of the right side: 0.295803989 Break this problem into two subproblems by setting (x + 0.5) equal to 0.295803989 and -0.295803989.Subproblem 1
x + 0.5 = 0.295803989 Simplifying x + 0.5 = 0.295803989 Reorder the terms: 0.5 + x = 0.295803989 Solving 0.5 + x = 0.295803989 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 = 0.295803989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.295803989 + -0.5 x = 0.295803989 + -0.5 Combine like terms: 0.295803989 + -0.5 = -0.204196011 x = -0.204196011 Simplifying x = -0.204196011Subproblem 2
x + 0.5 = -0.295803989 Simplifying x + 0.5 = -0.295803989 Reorder the terms: 0.5 + x = -0.295803989 Solving 0.5 + x = -0.295803989 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 = -0.295803989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.295803989 + -0.5 x = -0.295803989 + -0.5 Combine like terms: -0.295803989 + -0.5 = -0.795803989 x = -0.795803989 Simplifying x = -0.795803989Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.204196011, -0.795803989}Solution
x = {0, -0.204196011, -0.795803989}
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