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Simplifying (2x + -5)(2x + -5) = 23 Reorder the terms: (-5 + 2x)(2x + -5) = 23 Reorder the terms: (-5 + 2x)(-5 + 2x) = 23 Multiply (-5 + 2x) * (-5 + 2x) (-5(-5 + 2x) + 2x * (-5 + 2x)) = 23 ((-5 * -5 + 2x * -5) + 2x * (-5 + 2x)) = 23 ((25 + -10x) + 2x * (-5 + 2x)) = 23 (25 + -10x + (-5 * 2x + 2x * 2x)) = 23 (25 + -10x + (-10x + 4x2)) = 23 Combine like terms: -10x + -10x = -20x (25 + -20x + 4x2) = 23 Solving 25 + -20x + 4x2 = 23 Solving for variable 'x'. Reorder the terms: 25 + -23 + -20x + 4x2 = 23 + -23 Combine like terms: 25 + -23 = 2 2 + -20x + 4x2 = 23 + -23 Combine like terms: 23 + -23 = 0 2 + -20x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(1 + -10x + 2x2) = 0 Ignore the factor 2.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 = 4.897915762 x = 4.897915762 Simplifying x = 4.897915762Subproblem 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 = 0.102084238 x = 0.102084238 Simplifying x = 0.102084238Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.897915762, 0.102084238}Solution
x = {4.897915762, 0.102084238}
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