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Simplifying 19 = 2(x + 1) * x Reorder the terms: 19 = 2(1 + x) * x Reorder the terms for easier multiplication: 19 = 2x(1 + x) 19 = (1 * 2x + x * 2x) 19 = (2x + 2x2) Solving 19 = 2x + 2x2 Solving for variable 'x'. Reorder the terms: 19 + -2x + -2x2 = 2x + -2x + 2x2 + -2x2 Combine like terms: 2x + -2x = 0 19 + -2x + -2x2 = 0 + 2x2 + -2x2 19 + -2x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 19 + -2x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -9.5 + x + x2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + x + 9.5 + x2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + x + x2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + x + x2 = 0 + 9.5 x + x2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 x + x2 = 9.5 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. + 0.25 + x2 = 9.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 9.5 + 0.25 Combine like terms: 9.5 + 0.25 = 9.75 1.25 + x2 = 9.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 9.75 Calculate the square root of the right side: 3.122498999 Break this problem into two subproblems by setting (x + 0.5) equal to 3.122498999 and -3.122498999.Subproblem 1
x + 0.5 = 3.122498999 Simplifying x + 0.5 = 3.122498999 Reorder the terms: 0.5 + x = 3.122498999 Solving 0.5 + x = 3.122498999 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 = 3.122498999 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.122498999 + -0.5 x = 3.122498999 + -0.5 Combine like terms: 3.122498999 + -0.5 = 2.622498999 x = 2.622498999 Simplifying x = 2.622498999Subproblem 2
x + 0.5 = -3.122498999 Simplifying x + 0.5 = -3.122498999 Reorder the terms: 0.5 + x = -3.122498999 Solving 0.5 + x = -3.122498999 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 = -3.122498999 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.122498999 + -0.5 x = -3.122498999 + -0.5 Combine like terms: -3.122498999 + -0.5 = -3.622498999 x = -3.622498999 Simplifying x = -3.622498999Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.622498999, -3.622498999}
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