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Simplifying 5x(x + -1) = 68 + -2x + -2x Reorder the terms: 5x(-1 + x) = 68 + -2x + -2x (-1 * 5x + x * 5x) = 68 + -2x + -2x (-5x + 5x2) = 68 + -2x + -2x Combine like terms: -2x + -2x = -4x -5x + 5x2 = 68 + -4x Solving -5x + 5x2 = 68 + -4x Solving for variable 'x'. Reorder the terms: -68 + -5x + 4x + 5x2 = 68 + -4x + -68 + 4x Combine like terms: -5x + 4x = -1x -68 + -1x + 5x2 = 68 + -4x + -68 + 4x Reorder the terms: -68 + -1x + 5x2 = 68 + -68 + -4x + 4x Combine like terms: 68 + -68 = 0 -68 + -1x + 5x2 = 0 + -4x + 4x -68 + -1x + 5x2 = -4x + 4x Combine like terms: -4x + 4x = 0 -68 + -1x + 5x2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -13.6 + -0.2x + x2 = 0 Move the constant term to the right: Add '13.6' to each side of the equation. -13.6 + -0.2x + 13.6 + x2 = 0 + 13.6 Reorder the terms: -13.6 + 13.6 + -0.2x + x2 = 0 + 13.6 Combine like terms: -13.6 + 13.6 = 0.0 0.0 + -0.2x + x2 = 0 + 13.6 -0.2x + x2 = 0 + 13.6 Combine like terms: 0 + 13.6 = 13.6 -0.2x + x2 = 13.6 The x term is -0.2x. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2x + 0.01 + x2 = 13.6 + 0.01 Reorder the terms: 0.01 + -0.2x + x2 = 13.6 + 0.01 Combine like terms: 13.6 + 0.01 = 13.61 0.01 + -0.2x + x2 = 13.61 Factor a perfect square on the left side: (x + -0.1)(x + -0.1) = 13.61 Calculate the square root of the right side: 3.689173349 Break this problem into two subproblems by setting (x + -0.1) equal to 3.689173349 and -3.689173349.Subproblem 1
x + -0.1 = 3.689173349 Simplifying x + -0.1 = 3.689173349 Reorder the terms: -0.1 + x = 3.689173349 Solving -0.1 + x = 3.689173349 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = 3.689173349 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = 3.689173349 + 0.1 x = 3.689173349 + 0.1 Combine like terms: 3.689173349 + 0.1 = 3.789173349 x = 3.789173349 Simplifying x = 3.789173349Subproblem 2
x + -0.1 = -3.689173349 Simplifying x + -0.1 = -3.689173349 Reorder the terms: -0.1 + x = -3.689173349 Solving -0.1 + x = -3.689173349 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = -3.689173349 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = -3.689173349 + 0.1 x = -3.689173349 + 0.1 Combine like terms: -3.689173349 + 0.1 = -3.589173349 x = -3.589173349 Simplifying x = -3.589173349Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.789173349, -3.589173349}
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