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Simplifying (x + 0.5)(2x + -0.33) = 0 Reorder the terms: (0.5 + x)(2x + -0.33) = 0 Reorder the terms: (0.5 + x)(-0.33 + 2x) = 0 Multiply (0.5 + x) * (-0.33 + 2x) (0.5(-0.33 + 2x) + x(-0.33 + 2x)) = 0 ((-0.33 * 0.5 + 2x * 0.5) + x(-0.33 + 2x)) = 0 ((-0.165 + 1x) + x(-0.33 + 2x)) = 0 (-0.165 + 1x + (-0.33 * x + 2x * x)) = 0 (-0.165 + 1x + (-0.33x + 2x2)) = 0 Combine like terms: 1x + -0.33x = 0.67x (-0.165 + 0.67x + 2x2) = 0 Solving -0.165 + 0.67x + 2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.0825 + 0.335x + x2 = 0 Move the constant term to the right: Add '0.0825' to each side of the equation. -0.0825 + 0.335x + 0.0825 + x2 = 0 + 0.0825 Reorder the terms: -0.0825 + 0.0825 + 0.335x + x2 = 0 + 0.0825 Combine like terms: -0.0825 + 0.0825 = 0.0000 0.0000 + 0.335x + x2 = 0 + 0.0825 0.335x + x2 = 0 + 0.0825 Combine like terms: 0 + 0.0825 = 0.0825 0.335x + x2 = 0.0825 The x term is 0.335x. Take half its coefficient (0.1675). Square it (0.02805625) and add it to both sides. Add '0.02805625' to each side of the equation. 0.335x + 0.02805625 + x2 = 0.0825 + 0.02805625 Reorder the terms: 0.02805625 + 0.335x + x2 = 0.0825 + 0.02805625 Combine like terms: 0.0825 + 0.02805625 = 0.11055625 0.02805625 + 0.335x + x2 = 0.11055625 Factor a perfect square on the left side: (x + 0.1675)(x + 0.1675) = 0.11055625 Calculate the square root of the right side: 0.3325 Break this problem into two subproblems by setting (x + 0.1675) equal to 0.3325 and -0.3325.Subproblem 1
x + 0.1675 = 0.3325 Simplifying x + 0.1675 = 0.3325 Reorder the terms: 0.1675 + x = 0.3325 Solving 0.1675 + x = 0.3325 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1675' to each side of the equation. 0.1675 + -0.1675 + x = 0.3325 + -0.1675 Combine like terms: 0.1675 + -0.1675 = 0.0000 0.0000 + x = 0.3325 + -0.1675 x = 0.3325 + -0.1675 Combine like terms: 0.3325 + -0.1675 = 0.165 x = 0.165 Simplifying x = 0.165Subproblem 2
x + 0.1675 = -0.3325 Simplifying x + 0.1675 = -0.3325 Reorder the terms: 0.1675 + x = -0.3325 Solving 0.1675 + x = -0.3325 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1675' to each side of the equation. 0.1675 + -0.1675 + x = -0.3325 + -0.1675 Combine like terms: 0.1675 + -0.1675 = 0.0000 0.0000 + x = -0.3325 + -0.1675 x = -0.3325 + -0.1675 Combine like terms: -0.3325 + -0.1675 = -0.5 x = -0.5 Simplifying x = -0.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.165, -0.5}
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