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Simplifying x = 0.1x2 + -0.5x + 1.4 Reorder the terms: x = 1.4 + -0.5x + 0.1x2 Solving x = 1.4 + -0.5x + 0.1x2 Solving for variable 'x'. Reorder the terms: -1.4 + x + 0.5x + -0.1x2 = 1.4 + -0.5x + 0.1x2 + -1.4 + 0.5x + -0.1x2 Combine like terms: x + 0.5x = 1.5x -1.4 + 1.5x + -0.1x2 = 1.4 + -0.5x + 0.1x2 + -1.4 + 0.5x + -0.1x2 Reorder the terms: -1.4 + 1.5x + -0.1x2 = 1.4 + -1.4 + -0.5x + 0.5x + 0.1x2 + -0.1x2 Combine like terms: 1.4 + -1.4 = 0.0 -1.4 + 1.5x + -0.1x2 = 0.0 + -0.5x + 0.5x + 0.1x2 + -0.1x2 -1.4 + 1.5x + -0.1x2 = -0.5x + 0.5x + 0.1x2 + -0.1x2 Combine like terms: -0.5x + 0.5x = 0.0 -1.4 + 1.5x + -0.1x2 = 0.0 + 0.1x2 + -0.1x2 -1.4 + 1.5x + -0.1x2 = 0.1x2 + -0.1x2 Combine like terms: 0.1x2 + -0.1x2 = 0.0 -1.4 + 1.5x + -0.1x2 = 0.0 Begin completing the square. Divide all terms by -0.1 the coefficient of the squared term: Divide each side by '-0.1'. 14 + -15x + x2 = 0 Move the constant term to the right: Add '-14' to each side of the equation. 14 + -15x + -14 + x2 = 0 + -14 Reorder the terms: 14 + -14 + -15x + x2 = 0 + -14 Combine like terms: 14 + -14 = 0 0 + -15x + x2 = 0 + -14 -15x + x2 = 0 + -14 Combine like terms: 0 + -14 = -14 -15x + x2 = -14 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 = -14 + 56.25 Reorder the terms: 56.25 + -15x + x2 = -14 + 56.25 Combine like terms: -14 + 56.25 = 42.25 56.25 + -15x + x2 = 42.25 Factor a perfect square on the left side: (x + -7.5)(x + -7.5) = 42.25 Calculate the square root of the right side: 6.5 Break this problem into two subproblems by setting (x + -7.5) equal to 6.5 and -6.5.Subproblem 1
x + -7.5 = 6.5 Simplifying x + -7.5 = 6.5 Reorder the terms: -7.5 + x = 6.5 Solving -7.5 + x = 6.5 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 = 6.5 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = 6.5 + 7.5 x = 6.5 + 7.5 Combine like terms: 6.5 + 7.5 = 14 x = 14 Simplifying x = 14Subproblem 2
x + -7.5 = -6.5 Simplifying x + -7.5 = -6.5 Reorder the terms: -7.5 + x = -6.5 Solving -7.5 + x = -6.5 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 = -6.5 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = -6.5 + 7.5 x = -6.5 + 7.5 Combine like terms: -6.5 + 7.5 = 1 x = 1 Simplifying x = 1Solution
The solution to the problem is based on the solutions from the subproblems. x = {14, 1}
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