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Simplifying -2x2 = 10x + 1 Reorder the terms: -2x2 = 1 + 10x Solving -2x2 = 1 + 10x Solving for variable 'x'. Reorder the terms: -1 + -10x + -2x2 = 1 + 10x + -1 + -10x Reorder the terms: -1 + -10x + -2x2 = 1 + -1 + 10x + -10x Combine like terms: 1 + -1 = 0 -1 + -10x + -2x2 = 0 + 10x + -10x -1 + -10x + -2x2 = 10x + -10x Combine like terms: 10x + -10x = 0 -1 + -10x + -2x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + 10x + 2x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + 10x + 2x2)' equal to zero and attempt to solve: Simplifying 1 + 10x + 2x2 = 0 Solving 1 + 10x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + 5x + x2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 5x + -0.5 + x2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 5x + x2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 5x + x2 = 0 + -0.5 5x + x2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 5x + x2 = -0.5 The x term is 5x. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5x + 6.25 + x2 = -0.5 + 6.25 Reorder the terms: 6.25 + 5x + x2 = -0.5 + 6.25 Combine like terms: -0.5 + 6.25 = 5.75 6.25 + 5x + x2 = 5.75 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 5.75 Calculate the square root of the right side: 2.397915762 Break this problem into two subproblems by setting (x + 2.5) equal to 2.397915762 and -2.397915762.Subproblem 1
x + 2.5 = 2.397915762 Simplifying x + 2.5 = 2.397915762 Reorder the terms: 2.5 + x = 2.397915762 Solving 2.5 + x = 2.397915762 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 2.397915762 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 2.397915762 + -2.5 x = 2.397915762 + -2.5 Combine like terms: 2.397915762 + -2.5 = -0.102084238 x = -0.102084238 Simplifying x = -0.102084238Subproblem 2
x + 2.5 = -2.397915762 Simplifying x + 2.5 = -2.397915762 Reorder the terms: 2.5 + x = -2.397915762 Solving 2.5 + x = -2.397915762 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -2.397915762 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -2.397915762 + -2.5 x = -2.397915762 + -2.5 Combine like terms: -2.397915762 + -2.5 = -4.897915762 x = -4.897915762 Simplifying x = -4.897915762Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.102084238, -4.897915762}Solution
x = {-0.102084238, -4.897915762}
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