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Simplifying 500 = 1.36x2 + 27.8x + 30 Reorder the terms: 500 = 30 + 27.8x + 1.36x2 Solving 500 = 30 + 27.8x + 1.36x2 Solving for variable 'x'. Combine like terms: 500 + -30 = 470 470 + -27.8x + -1.36x2 = 30 + 27.8x + 1.36x2 + -30 + -27.8x + -1.36x2 Reorder the terms: 470 + -27.8x + -1.36x2 = 30 + -30 + 27.8x + -27.8x + 1.36x2 + -1.36x2 Combine like terms: 30 + -30 = 0 470 + -27.8x + -1.36x2 = 0 + 27.8x + -27.8x + 1.36x2 + -1.36x2 470 + -27.8x + -1.36x2 = 27.8x + -27.8x + 1.36x2 + -1.36x2 Combine like terms: 27.8x + -27.8x = 0.0 470 + -27.8x + -1.36x2 = 0.0 + 1.36x2 + -1.36x2 470 + -27.8x + -1.36x2 = 1.36x2 + -1.36x2 Combine like terms: 1.36x2 + -1.36x2 = 0.00 470 + -27.8x + -1.36x2 = 0.00 Begin completing the square. Divide all terms by -1.36 the coefficient of the squared term: Divide each side by '-1.36'. -345.5882353 + 20.44117647x + x2 = 0 Move the constant term to the right: Add '345.5882353' to each side of the equation. -345.5882353 + 20.44117647x + 345.5882353 + x2 = 0 + 345.5882353 Reorder the terms: -345.5882353 + 345.5882353 + 20.44117647x + x2 = 0 + 345.5882353 Combine like terms: -345.5882353 + 345.5882353 = 0.0000000 0.0000000 + 20.44117647x + x2 = 0 + 345.5882353 20.44117647x + x2 = 0 + 345.5882353 Combine like terms: 0 + 345.5882353 = 345.5882353 20.44117647x + x2 = 345.5882353 The x term is 20.44117647x. Take half its coefficient (10.22058824). Square it (104.4604240) and add it to both sides. Add '104.4604240' to each side of the equation. 20.44117647x + 104.4604240 + x2 = 345.5882353 + 104.4604240 Reorder the terms: 104.4604240 + 20.44117647x + x2 = 345.5882353 + 104.4604240 Combine like terms: 345.5882353 + 104.4604240 = 450.0486593 104.4604240 + 20.44117647x + x2 = 450.0486593 Factor a perfect square on the left side: (x + 10.22058824)(x + 10.22058824) = 450.0486593 Calculate the square root of the right side: 21.214350315 Break this problem into two subproblems by setting (x + 10.22058824) equal to 21.214350315 and -21.214350315.Subproblem 1
x + 10.22058824 = 21.214350315 Simplifying x + 10.22058824 = 21.214350315 Reorder the terms: 10.22058824 + x = 21.214350315 Solving 10.22058824 + x = 21.214350315 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.22058824' to each side of the equation. 10.22058824 + -10.22058824 + x = 21.214350315 + -10.22058824 Combine like terms: 10.22058824 + -10.22058824 = 0.00000000 0.00000000 + x = 21.214350315 + -10.22058824 x = 21.214350315 + -10.22058824 Combine like terms: 21.214350315 + -10.22058824 = 10.993762075 x = 10.993762075 Simplifying x = 10.993762075Subproblem 2
x + 10.22058824 = -21.214350315 Simplifying x + 10.22058824 = -21.214350315 Reorder the terms: 10.22058824 + x = -21.214350315 Solving 10.22058824 + x = -21.214350315 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.22058824' to each side of the equation. 10.22058824 + -10.22058824 + x = -21.214350315 + -10.22058824 Combine like terms: 10.22058824 + -10.22058824 = 0.00000000 0.00000000 + x = -21.214350315 + -10.22058824 x = -21.214350315 + -10.22058824 Combine like terms: -21.214350315 + -10.22058824 = -31.434938555 x = -31.434938555 Simplifying x = -31.434938555Solution
The solution to the problem is based on the solutions from the subproblems. x = {10.993762075, -31.434938555}
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