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Simplifying 2x + 1 = x2 + 2.6x + 0.45 Reorder the terms: 1 + 2x = x2 + 2.6x + 0.45 Reorder the terms: 1 + 2x = 0.45 + 2.6x + x2 Solving 1 + 2x = 0.45 + 2.6x + x2 Solving for variable 'x'. Reorder the terms: 1 + -0.45 + 2x + -2.6x + -1x2 = 0.45 + 2.6x + x2 + -0.45 + -2.6x + -1x2 Combine like terms: 1 + -0.45 = 0.55 0.55 + 2x + -2.6x + -1x2 = 0.45 + 2.6x + x2 + -0.45 + -2.6x + -1x2 Combine like terms: 2x + -2.6x = -0.6x 0.55 + -0.6x + -1x2 = 0.45 + 2.6x + x2 + -0.45 + -2.6x + -1x2 Reorder the terms: 0.55 + -0.6x + -1x2 = 0.45 + -0.45 + 2.6x + -2.6x + x2 + -1x2 Combine like terms: 0.45 + -0.45 = 0.00 0.55 + -0.6x + -1x2 = 0.00 + 2.6x + -2.6x + x2 + -1x2 0.55 + -0.6x + -1x2 = 2.6x + -2.6x + x2 + -1x2 Combine like terms: 2.6x + -2.6x = 0.0 0.55 + -0.6x + -1x2 = 0.0 + x2 + -1x2 0.55 + -0.6x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 0.55 + -0.6x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -0.55 + 0.6x + x2 = 0 Move the constant term to the right: Add '0.55' to each side of the equation. -0.55 + 0.6x + 0.55 + x2 = 0 + 0.55 Reorder the terms: -0.55 + 0.55 + 0.6x + x2 = 0 + 0.55 Combine like terms: -0.55 + 0.55 = 0.00 0.00 + 0.6x + x2 = 0 + 0.55 0.6x + x2 = 0 + 0.55 Combine like terms: 0 + 0.55 = 0.55 0.6x + x2 = 0.55 The x term is 0.6x. Take half its coefficient (0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. 0.6x + 0.09 + x2 = 0.55 + 0.09 Reorder the terms: 0.09 + 0.6x + x2 = 0.55 + 0.09 Combine like terms: 0.55 + 0.09 = 0.64 0.09 + 0.6x + x2 = 0.64 Factor a perfect square on the left side: (x + 0.3)(x + 0.3) = 0.64 Calculate the square root of the right side: 0.8 Break this problem into two subproblems by setting (x + 0.3) equal to 0.8 and -0.8.Subproblem 1
x + 0.3 = 0.8 Simplifying x + 0.3 = 0.8 Reorder the terms: 0.3 + x = 0.8 Solving 0.3 + x = 0.8 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + x = 0.8 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + x = 0.8 + -0.3 x = 0.8 + -0.3 Combine like terms: 0.8 + -0.3 = 0.5 x = 0.5 Simplifying x = 0.5Subproblem 2
x + 0.3 = -0.8 Simplifying x + 0.3 = -0.8 Reorder the terms: 0.3 + x = -0.8 Solving 0.3 + x = -0.8 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + x = -0.8 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + x = -0.8 + -0.3 x = -0.8 + -0.3 Combine like terms: -0.8 + -0.3 = -1.1 x = -1.1 Simplifying x = -1.1Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.5, -1.1}
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