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Simplifying (0.5x + 6)(5x + -0.5) = 0 Reorder the terms: (6 + 0.5x)(5x + -0.5) = 0 Reorder the terms: (6 + 0.5x)(-0.5 + 5x) = 0 Multiply (6 + 0.5x) * (-0.5 + 5x) (6(-0.5 + 5x) + 0.5x * (-0.5 + 5x)) = 0 ((-0.5 * 6 + 5x * 6) + 0.5x * (-0.5 + 5x)) = 0 ((-3 + 30x) + 0.5x * (-0.5 + 5x)) = 0 (-3 + 30x + (-0.5 * 0.5x + 5x * 0.5x)) = 0 (-3 + 30x + (-0.25x + 2.5x2)) = 0 Combine like terms: 30x + -0.25x = 29.75x (-3 + 29.75x + 2.5x2) = 0 Solving -3 + 29.75x + 2.5x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2.5 the coefficient of the squared term: Divide each side by '2.5'. -1.2 + 11.9x + x2 = 0 Move the constant term to the right: Add '1.2' to each side of the equation. -1.2 + 11.9x + 1.2 + x2 = 0 + 1.2 Reorder the terms: -1.2 + 1.2 + 11.9x + x2 = 0 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + 11.9x + x2 = 0 + 1.2 11.9x + x2 = 0 + 1.2 Combine like terms: 0 + 1.2 = 1.2 11.9x + x2 = 1.2 The x term is 11.9x. Take half its coefficient (5.95). Square it (35.4025) and add it to both sides. Add '35.4025' to each side of the equation. 11.9x + 35.4025 + x2 = 1.2 + 35.4025 Reorder the terms: 35.4025 + 11.9x + x2 = 1.2 + 35.4025 Combine like terms: 1.2 + 35.4025 = 36.6025 35.4025 + 11.9x + x2 = 36.6025 Factor a perfect square on the left side: (x + 5.95)(x + 5.95) = 36.6025 Calculate the square root of the right side: 6.05 Break this problem into two subproblems by setting (x + 5.95) equal to 6.05 and -6.05.Subproblem 1
x + 5.95 = 6.05 Simplifying x + 5.95 = 6.05 Reorder the terms: 5.95 + x = 6.05 Solving 5.95 + x = 6.05 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.95' to each side of the equation. 5.95 + -5.95 + x = 6.05 + -5.95 Combine like terms: 5.95 + -5.95 = 0.00 0.00 + x = 6.05 + -5.95 x = 6.05 + -5.95 Combine like terms: 6.05 + -5.95 = 0.1 x = 0.1 Simplifying x = 0.1Subproblem 2
x + 5.95 = -6.05 Simplifying x + 5.95 = -6.05 Reorder the terms: 5.95 + x = -6.05 Solving 5.95 + x = -6.05 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.95' to each side of the equation. 5.95 + -5.95 + x = -6.05 + -5.95 Combine like terms: 5.95 + -5.95 = 0.00 0.00 + x = -6.05 + -5.95 x = -6.05 + -5.95 Combine like terms: -6.05 + -5.95 = -12 x = -12 Simplifying x = -12Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.1, -12}
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