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Simplifying 3(-6x + 9) * 10x = 1 Reorder the terms: 3(9 + -6x) * 10x = 1 Reorder the terms for easier multiplication: 3 * 10x(9 + -6x) = 1 Multiply 3 * 10 30x(9 + -6x) = 1 (9 * 30x + -6x * 30x) = 1 (270x + -180x2) = 1 Solving 270x + -180x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + 270x + -180x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 270x + -180x2 = 0 Begin completing the square. Divide all terms by -180 the coefficient of the squared term: Divide each side by '-180'. 0.005555555556 + -1.5x + x2 = 0.0 Move the constant term to the right: Add '-0.005555555556' to each side of the equation. 0.005555555556 + -1.5x + -0.005555555556 + x2 = 0.0 + -0.005555555556 Reorder the terms: 0.005555555556 + -0.005555555556 + -1.5x + x2 = 0.0 + -0.005555555556 Combine like terms: 0.005555555556 + -0.005555555556 = 0.000000000000 0.000000000000 + -1.5x + x2 = 0.0 + -0.005555555556 -1.5x + x2 = 0.0 + -0.005555555556 Combine like terms: 0.0 + -0.005555555556 = -0.005555555556 -1.5x + x2 = -0.005555555556 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = -0.005555555556 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = -0.005555555556 + 0.5625 Combine like terms: -0.005555555556 + 0.5625 = 0.556944444444 0.5625 + -1.5x + x2 = 0.556944444444 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 0.556944444444 Calculate the square root of the right side: 0.746287106 Break this problem into two subproblems by setting (x + -0.75) equal to 0.746287106 and -0.746287106.Subproblem 1
x + -0.75 = 0.746287106 Simplifying x + -0.75 = 0.746287106 Reorder the terms: -0.75 + x = 0.746287106 Solving -0.75 + x = 0.746287106 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 0.746287106 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 0.746287106 + 0.75 x = 0.746287106 + 0.75 Combine like terms: 0.746287106 + 0.75 = 1.496287106 x = 1.496287106 Simplifying x = 1.496287106Subproblem 2
x + -0.75 = -0.746287106 Simplifying x + -0.75 = -0.746287106 Reorder the terms: -0.75 + x = -0.746287106 Solving -0.75 + x = -0.746287106 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -0.746287106 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -0.746287106 + 0.75 x = -0.746287106 + 0.75 Combine like terms: -0.746287106 + 0.75 = 0.003712894 x = 0.003712894 Simplifying x = 0.003712894Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.496287106, 0.003712894}
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