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Simplifying 2(x + -0.1)(2x + -1.5) = 0 Reorder the terms: 2(-0.1 + x)(2x + -1.5) = 0 Reorder the terms: 2(-0.1 + x)(-1.5 + 2x) = 0 Multiply (-0.1 + x) * (-1.5 + 2x) 2(-0.1(-1.5 + 2x) + x(-1.5 + 2x)) = 0 2((-1.5 * -0.1 + 2x * -0.1) + x(-1.5 + 2x)) = 0 2((0.15 + -0.2x) + x(-1.5 + 2x)) = 0 2(0.15 + -0.2x + (-1.5 * x + 2x * x)) = 0 2(0.15 + -0.2x + (-1.5x + 2x2)) = 0 Combine like terms: -0.2x + -1.5x = -1.7x 2(0.15 + -1.7x + 2x2) = 0 (0.15 * 2 + -1.7x * 2 + 2x2 * 2) = 0 (0.3 + -3.4x + 4x2) = 0 Solving 0.3 + -3.4x + 4x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.075 + -0.85x + x2 = 0 Move the constant term to the right: Add '-0.075' to each side of the equation. 0.075 + -0.85x + -0.075 + x2 = 0 + -0.075 Reorder the terms: 0.075 + -0.075 + -0.85x + x2 = 0 + -0.075 Combine like terms: 0.075 + -0.075 = 0.000 0.000 + -0.85x + x2 = 0 + -0.075 -0.85x + x2 = 0 + -0.075 Combine like terms: 0 + -0.075 = -0.075 -0.85x + x2 = -0.075 The x term is -0.85x. Take half its coefficient (-0.425). Square it (0.180625) and add it to both sides. Add '0.180625' to each side of the equation. -0.85x + 0.180625 + x2 = -0.075 + 0.180625 Reorder the terms: 0.180625 + -0.85x + x2 = -0.075 + 0.180625 Combine like terms: -0.075 + 0.180625 = 0.105625 0.180625 + -0.85x + x2 = 0.105625 Factor a perfect square on the left side: (x + -0.425)(x + -0.425) = 0.105625 Calculate the square root of the right side: 0.325 Break this problem into two subproblems by setting (x + -0.425) equal to 0.325 and -0.325.Subproblem 1
x + -0.425 = 0.325 Simplifying x + -0.425 = 0.325 Reorder the terms: -0.425 + x = 0.325 Solving -0.425 + x = 0.325 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.425' to each side of the equation. -0.425 + 0.425 + x = 0.325 + 0.425 Combine like terms: -0.425 + 0.425 = 0.000 0.000 + x = 0.325 + 0.425 x = 0.325 + 0.425 Combine like terms: 0.325 + 0.425 = 0.75 x = 0.75 Simplifying x = 0.75Subproblem 2
x + -0.425 = -0.325 Simplifying x + -0.425 = -0.325 Reorder the terms: -0.425 + x = -0.325 Solving -0.425 + x = -0.325 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.425' to each side of the equation. -0.425 + 0.425 + x = -0.325 + 0.425 Combine like terms: -0.425 + 0.425 = 0.000 0.000 + x = -0.325 + 0.425 x = -0.325 + 0.425 Combine like terms: -0.325 + 0.425 = 0.1 x = 0.1 Simplifying x = 0.1Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.75, 0.1}
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