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Simplifying 5x(3x + 3) * x + 6 = x + 6 Reorder the terms: 5x(3 + 3x) * x + 6 = x + 6 Reorder the terms for easier multiplication: 5x * x(3 + 3x) + 6 = x + 6 Multiply x * x 5x2(3 + 3x) + 6 = x + 6 (3 * 5x2 + 3x * 5x2) + 6 = x + 6 (15x2 + 15x3) + 6 = x + 6 Reorder the terms: 6 + 15x2 + 15x3 = x + 6 Reorder the terms: 6 + 15x2 + 15x3 = 6 + x Add '-6' to each side of the equation. 6 + 15x2 + -6 + 15x3 = 6 + -6 + x Reorder the terms: 6 + -6 + 15x2 + 15x3 = 6 + -6 + x Combine like terms: 6 + -6 = 0 0 + 15x2 + 15x3 = 6 + -6 + x 15x2 + 15x3 = 6 + -6 + x Combine like terms: 6 + -6 = 0 15x2 + 15x3 = 0 + x 15x2 + 15x3 = x Solving 15x2 + 15x3 = x Solving for variable 'x'. Reorder the terms: -1x + 15x2 + 15x3 = x + -1x Combine like terms: x + -1x = 0 -1x + 15x2 + 15x3 = 0 Factor out the Greatest Common Factor (GCF), 'x'. x(-1 + 15x + 15x2) = 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 '(-1 + 15x + 15x2)' equal to zero and attempt to solve: Simplifying -1 + 15x + 15x2 = 0 Solving -1 + 15x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.06666666667 + x + x2 = 0 Move the constant term to the right: Add '0.06666666667' to each side of the equation. -0.06666666667 + x + 0.06666666667 + x2 = 0 + 0.06666666667 Reorder the terms: -0.06666666667 + 0.06666666667 + x + x2 = 0 + 0.06666666667 Combine like terms: -0.06666666667 + 0.06666666667 = 0.00000000000 0.00000000000 + x + x2 = 0 + 0.06666666667 x + x2 = 0 + 0.06666666667 Combine like terms: 0 + 0.06666666667 = 0.06666666667 x + x2 = 0.06666666667 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. + 0.25 + x2 = 0.06666666667 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 0.06666666667 + 0.25 Combine like terms: 0.06666666667 + 0.25 = 0.31666666667 1.25 + x2 = 0.31666666667 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.31666666667 Calculate the square root of the right side: 0.562731434 Break this problem into two subproblems by setting (x + 0.5) equal to 0.562731434 and -0.562731434.Subproblem 1
x + 0.5 = 0.562731434 Simplifying x + 0.5 = 0.562731434 Reorder the terms: 0.5 + x = 0.562731434 Solving 0.5 + x = 0.562731434 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 = 0.562731434 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.562731434 + -0.5 x = 0.562731434 + -0.5 Combine like terms: 0.562731434 + -0.5 = 0.062731434 x = 0.062731434 Simplifying x = 0.062731434Subproblem 2
x + 0.5 = -0.562731434 Simplifying x + 0.5 = -0.562731434 Reorder the terms: 0.5 + x = -0.562731434 Solving 0.5 + x = -0.562731434 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 = -0.562731434 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.562731434 + -0.5 x = -0.562731434 + -0.5 Combine like terms: -0.562731434 + -0.5 = -1.062731434 x = -1.062731434 Simplifying x = -1.062731434Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.062731434, -1.062731434}Solution
x = {0, 0.062731434, -1.062731434}
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