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Simplifying 6x(4x2 + 1) + -15x2(2x + -1) = 0 Reorder the terms: 6x(1 + 4x2) + -15x2(2x + -1) = 0 (1 * 6x + 4x2 * 6x) + -15x2(2x + -1) = 0 (6x + 24x3) + -15x2(2x + -1) = 0 Reorder the terms: 6x + 24x3 + -15x2(-1 + 2x) = 0 6x + 24x3 + (-1 * -15x2 + 2x * -15x2) = 0 6x + 24x3 + (15x2 + -30x3) = 0 Reorder the terms: 6x + 15x2 + 24x3 + -30x3 = 0 Combine like terms: 24x3 + -30x3 = -6x3 6x + 15x2 + -6x3 = 0 Solving 6x + 15x2 + -6x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3x'. 3x(2 + 5x + -2x2) = 0 Ignore the factor 3.Subproblem 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 '(2 + 5x + -2x2)' equal to zero and attempt to solve: Simplifying 2 + 5x + -2x2 = 0 Solving 2 + 5x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1 + -2.5x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -2.5x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + -2.5x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2.5x + x2 = 0 + 1 -2.5x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 -2.5x + x2 = 1 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = 1 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = 1 + 1.5625 Combine like terms: 1 + 1.5625 = 2.5625 1.5625 + -2.5x + x2 = 2.5625 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (x + -1.25) equal to 1.600781059 and -1.600781059.Subproblem 1
x + -1.25 = 1.600781059 Simplifying x + -1.25 = 1.600781059 Reorder the terms: -1.25 + x = 1.600781059 Solving -1.25 + x = 1.600781059 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 1.600781059 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 1.600781059 + 1.25 x = 1.600781059 + 1.25 Combine like terms: 1.600781059 + 1.25 = 2.850781059 x = 2.850781059 Simplifying x = 2.850781059Subproblem 2
x + -1.25 = -1.600781059 Simplifying x + -1.25 = -1.600781059 Reorder the terms: -1.25 + x = -1.600781059 Solving -1.25 + x = -1.600781059 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -1.600781059 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -1.600781059 + 1.25 x = -1.600781059 + 1.25 Combine like terms: -1.600781059 + 1.25 = -0.350781059 x = -0.350781059 Simplifying x = -0.350781059Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.850781059, -0.350781059}Solution
x = {0, 2.850781059, -0.350781059}
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