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Simplifying 1.3x + 2(0.6x + -4) = 3x + -0.5x(x + 16) Reorder the terms: 1.3x + 2(-4 + 0.6x) = 3x + -0.5x(x + 16) 1.3x + (-4 * 2 + 0.6x * 2) = 3x + -0.5x(x + 16) 1.3x + (-8 + 1.2x) = 3x + -0.5x(x + 16) Reorder the terms: -8 + 1.3x + 1.2x = 3x + -0.5x(x + 16) Combine like terms: 1.3x + 1.2x = 2.5x -8 + 2.5x = 3x + -0.5x(x + 16) Reorder the terms: -8 + 2.5x = 3x + -0.5x(16 + x) -8 + 2.5x = 3x + (16 * -0.5x + x * -0.5x) -8 + 2.5x = 3x + (-8x + -0.5x2) Combine like terms: 3x + -8x = -5x -8 + 2.5x = -5x + -0.5x2 Solving -8 + 2.5x = -5x + -0.5x2 Solving for variable 'x'. Combine like terms: 2.5x + 5x = 7.5x -8 + 7.5x + 0.5x2 = -5x + -0.5x2 + 5x + 0.5x2 Reorder the terms: -8 + 7.5x + 0.5x2 = -5x + 5x + -0.5x2 + 0.5x2 Combine like terms: -5x + 5x = 0 -8 + 7.5x + 0.5x2 = 0 + -0.5x2 + 0.5x2 -8 + 7.5x + 0.5x2 = -0.5x2 + 0.5x2 Combine like terms: -0.5x2 + 0.5x2 = 0.0 -8 + 7.5x + 0.5x2 = 0.0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -16 + 15x + x2 = 0 Move the constant term to the right: Add '16' to each side of the equation. -16 + 15x + 16 + x2 = 0 + 16 Reorder the terms: -16 + 16 + 15x + x2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + 15x + x2 = 0 + 16 15x + x2 = 0 + 16 Combine like terms: 0 + 16 = 16 15x + x2 = 16 The x term is 15x. Take half its coefficient (7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. 15x + 56.25 + x2 = 16 + 56.25 Reorder the terms: 56.25 + 15x + x2 = 16 + 56.25 Combine like terms: 16 + 56.25 = 72.25 56.25 + 15x + x2 = 72.25 Factor a perfect square on the left side: (x + 7.5)(x + 7.5) = 72.25 Calculate the square root of the right side: 8.5 Break this problem into two subproblems by setting (x + 7.5) equal to 8.5 and -8.5.Subproblem 1
x + 7.5 = 8.5 Simplifying x + 7.5 = 8.5 Reorder the terms: 7.5 + x = 8.5 Solving 7.5 + x = 8.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = 8.5 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = 8.5 + -7.5 x = 8.5 + -7.5 Combine like terms: 8.5 + -7.5 = 1 x = 1 Simplifying x = 1Subproblem 2
x + 7.5 = -8.5 Simplifying x + 7.5 = -8.5 Reorder the terms: 7.5 + x = -8.5 Solving 7.5 + x = -8.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = -8.5 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = -8.5 + -7.5 x = -8.5 + -7.5 Combine like terms: -8.5 + -7.5 = -16 x = -16 Simplifying x = -16Solution
The solution to the problem is based on the solutions from the subproblems. x = {1, -16}
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