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Simplifying 10x + 10x(x + -1) + 5(x + -1) = 160 Reorder the terms: 10x + 10x(-1 + x) + 5(x + -1) = 160 10x + (-1 * 10x + x * 10x) + 5(x + -1) = 160 10x + (-10x + 10x2) + 5(x + -1) = 160 Reorder the terms: 10x + -10x + 10x2 + 5(-1 + x) = 160 10x + -10x + 10x2 + (-1 * 5 + x * 5) = 160 10x + -10x + 10x2 + (-5 + 5x) = 160 Reorder the terms: -5 + 10x + -10x + 5x + 10x2 = 160 Combine like terms: 10x + -10x = 0 -5 + 0 + 5x + 10x2 = 160 -5 + 5x + 10x2 = 160 Solving -5 + 5x + 10x2 = 160 Solving for variable 'x'. Reorder the terms: -5 + -160 + 5x + 10x2 = 160 + -160 Combine like terms: -5 + -160 = -165 -165 + 5x + 10x2 = 160 + -160 Combine like terms: 160 + -160 = 0 -165 + 5x + 10x2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-33 + x + 2x2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-33 + x + 2x2)' equal to zero and attempt to solve: Simplifying -33 + x + 2x2 = 0 Solving -33 + x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -16.5 + 0.5x + x2 = 0 Move the constant term to the right: Add '16.5' to each side of the equation. -16.5 + 0.5x + 16.5 + x2 = 0 + 16.5 Reorder the terms: -16.5 + 16.5 + 0.5x + x2 = 0 + 16.5 Combine like terms: -16.5 + 16.5 = 0.0 0.0 + 0.5x + x2 = 0 + 16.5 0.5x + x2 = 0 + 16.5 Combine like terms: 0 + 16.5 = 16.5 0.5x + x2 = 16.5 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5x + 0.25 + x2 = 16.5 + 0.25 Reorder the terms: 0.25 + 0.5x + x2 = 16.5 + 0.25 Combine like terms: 16.5 + 0.25 = 16.75 0.25 + 0.5x + x2 = 16.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 16.75 Calculate the square root of the right side: 4.092676386 Break this problem into two subproblems by setting (x + 0.5) equal to 4.092676386 and -4.092676386.Subproblem 1
x + 0.5 = 4.092676386 Simplifying x + 0.5 = 4.092676386 Reorder the terms: 0.5 + x = 4.092676386 Solving 0.5 + x = 4.092676386 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 4.092676386 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 4.092676386 + -0.5 x = 4.092676386 + -0.5 Combine like terms: 4.092676386 + -0.5 = 3.592676386 x = 3.592676386 Simplifying x = 3.592676386Subproblem 2
x + 0.5 = -4.092676386 Simplifying x + 0.5 = -4.092676386 Reorder the terms: 0.5 + x = -4.092676386 Solving 0.5 + x = -4.092676386 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -4.092676386 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -4.092676386 + -0.5 x = -4.092676386 + -0.5 Combine like terms: -4.092676386 + -0.5 = -4.592676386 x = -4.592676386 Simplifying x = -4.592676386Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.592676386, -4.592676386}Solution
x = {3.592676386, -4.592676386}
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