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Simplifying (4x + -0.5)(4 + x) = 0 Reorder the terms: (-0.5 + 4x)(4 + x) = 0 Multiply (-0.5 + 4x) * (4 + x) (-0.5(4 + x) + 4x * (4 + x)) = 0 ((4 * -0.5 + x * -0.5) + 4x * (4 + x)) = 0 ((-2 + -0.5x) + 4x * (4 + x)) = 0 (-2 + -0.5x + (4 * 4x + x * 4x)) = 0 (-2 + -0.5x + (16x + 4x2)) = 0 Combine like terms: -0.5x + 16x = 15.5x (-2 + 15.5x + 4x2) = 0 Solving -2 + 15.5x + 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'. -0.5 + 3.875x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 3.875x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 3.875x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 3.875x + x2 = 0 + 0.5 3.875x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 3.875x + x2 = 0.5 The x term is 3.875x. Take half its coefficient (1.9375). Square it (3.75390625) and add it to both sides. Add '3.75390625' to each side of the equation. 3.875x + 3.75390625 + x2 = 0.5 + 3.75390625 Reorder the terms: 3.75390625 + 3.875x + x2 = 0.5 + 3.75390625 Combine like terms: 0.5 + 3.75390625 = 4.25390625 3.75390625 + 3.875x + x2 = 4.25390625 Factor a perfect square on the left side: (x + 1.9375)(x + 1.9375) = 4.25390625 Calculate the square root of the right side: 2.0625 Break this problem into two subproblems by setting (x + 1.9375) equal to 2.0625 and -2.0625.Subproblem 1
x + 1.9375 = 2.0625 Simplifying x + 1.9375 = 2.0625 Reorder the terms: 1.9375 + x = 2.0625 Solving 1.9375 + x = 2.0625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.9375' to each side of the equation. 1.9375 + -1.9375 + x = 2.0625 + -1.9375 Combine like terms: 1.9375 + -1.9375 = 0.0000 0.0000 + x = 2.0625 + -1.9375 x = 2.0625 + -1.9375 Combine like terms: 2.0625 + -1.9375 = 0.125 x = 0.125 Simplifying x = 0.125Subproblem 2
x + 1.9375 = -2.0625 Simplifying x + 1.9375 = -2.0625 Reorder the terms: 1.9375 + x = -2.0625 Solving 1.9375 + x = -2.0625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.9375' to each side of the equation. 1.9375 + -1.9375 + x = -2.0625 + -1.9375 Combine like terms: 1.9375 + -1.9375 = 0.0000 0.0000 + x = -2.0625 + -1.9375 x = -2.0625 + -1.9375 Combine like terms: -2.0625 + -1.9375 = -4 x = -4 Simplifying x = -4Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.125, -4}
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