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Simplifying (0.8 + 2x)(0.25 + 2x) = 2.25 Multiply (0.8 + 2x) * (0.25 + 2x) (0.8(0.25 + 2x) + 2x * (0.25 + 2x)) = 2.25 ((0.25 * 0.8 + 2x * 0.8) + 2x * (0.25 + 2x)) = 2.25 ((0.2 + 1.6x) + 2x * (0.25 + 2x)) = 2.25 (0.2 + 1.6x + (0.25 * 2x + 2x * 2x)) = 2.25 (0.2 + 1.6x + (0.5x + 4x2)) = 2.25 Combine like terms: 1.6x + 0.5x = 2.1x (0.2 + 2.1x + 4x2) = 2.25 Solving 0.2 + 2.1x + 4x2 = 2.25 Solving for variable 'x'. Reorder the terms: 0.2 + -2.25 + 2.1x + 4x2 = 2.25 + -2.25 Combine like terms: 0.2 + -2.25 = -2.05 -2.05 + 2.1x + 4x2 = 2.25 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 -2.05 + 2.1x + 4x2 = 0.00 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.5125 + 0.525x + x2 = 0.00 Move the constant term to the right: Add '0.5125' to each side of the equation. -0.5125 + 0.525x + 0.5125 + x2 = 0.00 + 0.5125 Reorder the terms: -0.5125 + 0.5125 + 0.525x + x2 = 0.00 + 0.5125 Combine like terms: -0.5125 + 0.5125 = 0.0000 0.0000 + 0.525x + x2 = 0.00 + 0.5125 0.525x + x2 = 0.00 + 0.5125 Combine like terms: 0.00 + 0.5125 = 0.5125 0.525x + x2 = 0.5125 The x term is 0.525x. Take half its coefficient (0.2625). Square it (0.06890625) and add it to both sides. Add '0.06890625' to each side of the equation. 0.525x + 0.06890625 + x2 = 0.5125 + 0.06890625 Reorder the terms: 0.06890625 + 0.525x + x2 = 0.5125 + 0.06890625 Combine like terms: 0.5125 + 0.06890625 = 0.58140625 0.06890625 + 0.525x + x2 = 0.58140625 Factor a perfect square on the left side: (x + 0.2625)(x + 0.2625) = 0.58140625 Calculate the square root of the right side: 0.7625 Break this problem into two subproblems by setting (x + 0.2625) equal to 0.7625 and -0.7625.Subproblem 1
x + 0.2625 = 0.7625 Simplifying x + 0.2625 = 0.7625 Reorder the terms: 0.2625 + x = 0.7625 Solving 0.2625 + x = 0.7625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2625' to each side of the equation. 0.2625 + -0.2625 + x = 0.7625 + -0.2625 Combine like terms: 0.2625 + -0.2625 = 0.0000 0.0000 + x = 0.7625 + -0.2625 x = 0.7625 + -0.2625 Combine like terms: 0.7625 + -0.2625 = 0.5 x = 0.5 Simplifying x = 0.5Subproblem 2
x + 0.2625 = -0.7625 Simplifying x + 0.2625 = -0.7625 Reorder the terms: 0.2625 + x = -0.7625 Solving 0.2625 + x = -0.7625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2625' to each side of the equation. 0.2625 + -0.2625 + x = -0.7625 + -0.2625 Combine like terms: 0.2625 + -0.2625 = 0.0000 0.0000 + x = -0.7625 + -0.2625 x = -0.7625 + -0.2625 Combine like terms: -0.7625 + -0.2625 = -1.025 x = -1.025 Simplifying x = -1.025Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.5, -1.025}
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