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Simplifying 0.45(x2 + -2x + 1) = x2 Reorder the terms: 0.45(1 + -2x + x2) = x2 (1 * 0.45 + -2x * 0.45 + x2 * 0.45) = x2 (0.45 + -0.9x + 0.45x2) = x2 Solving 0.45 + -0.9x + 0.45x2 = x2 Solving for variable 'x'. Combine like terms: 0.45x2 + -1x2 = -0.55x2 0.45 + -0.9x + -0.55x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 0.45 + -0.9x + -0.55x2 = 0 Begin completing the square. Divide all terms by -0.55 the coefficient of the squared term: Divide each side by '-0.55'. -0.8181818182 + 1.636363636x + x2 = 0 Move the constant term to the right: Add '0.8181818182' to each side of the equation. -0.8181818182 + 1.636363636x + 0.8181818182 + x2 = 0 + 0.8181818182 Reorder the terms: -0.8181818182 + 0.8181818182 + 1.636363636x + x2 = 0 + 0.8181818182 Combine like terms: -0.8181818182 + 0.8181818182 = 0.0000000000 0.0000000000 + 1.636363636x + x2 = 0 + 0.8181818182 1.636363636x + x2 = 0 + 0.8181818182 Combine like terms: 0 + 0.8181818182 = 0.8181818182 1.636363636x + x2 = 0.8181818182 The x term is 1.636363636x. Take half its coefficient (0.818181818). Square it (0.6694214873) and add it to both sides. Add '0.6694214873' to each side of the equation. 1.636363636x + 0.6694214873 + x2 = 0.8181818182 + 0.6694214873 Reorder the terms: 0.6694214873 + 1.636363636x + x2 = 0.8181818182 + 0.6694214873 Combine like terms: 0.8181818182 + 0.6694214873 = 1.4876033055 0.6694214873 + 1.636363636x + x2 = 1.4876033055 Factor a perfect square on the left side: (x + 0.818181818)(x + 0.818181818) = 1.4876033055 Calculate the square root of the right side: 1.219673442 Break this problem into two subproblems by setting (x + 0.818181818) equal to 1.219673442 and -1.219673442.Subproblem 1
x + 0.818181818 = 1.219673442 Simplifying x + 0.818181818 = 1.219673442 Reorder the terms: 0.818181818 + x = 1.219673442 Solving 0.818181818 + x = 1.219673442 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.818181818' to each side of the equation. 0.818181818 + -0.818181818 + x = 1.219673442 + -0.818181818 Combine like terms: 0.818181818 + -0.818181818 = 0.000000000 0.000000000 + x = 1.219673442 + -0.818181818 x = 1.219673442 + -0.818181818 Combine like terms: 1.219673442 + -0.818181818 = 0.401491624 x = 0.401491624 Simplifying x = 0.401491624Subproblem 2
x + 0.818181818 = -1.219673442 Simplifying x + 0.818181818 = -1.219673442 Reorder the terms: 0.818181818 + x = -1.219673442 Solving 0.818181818 + x = -1.219673442 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.818181818' to each side of the equation. 0.818181818 + -0.818181818 + x = -1.219673442 + -0.818181818 Combine like terms: 0.818181818 + -0.818181818 = 0.000000000 0.000000000 + x = -1.219673442 + -0.818181818 x = -1.219673442 + -0.818181818 Combine like terms: -1.219673442 + -0.818181818 = -2.03785526 x = -2.03785526 Simplifying x = -2.03785526Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.401491624, -2.03785526}
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