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Simplifying 105 = 11x + 0.45x2 Solving 105 = 11x + 0.45x2 Solving for variable 'x'. Reorder the terms: 105 + -11x + -0.45x2 = 11x + -11x + 0.45x2 + -0.45x2 Combine like terms: 11x + -11x = 0 105 + -11x + -0.45x2 = 0 + 0.45x2 + -0.45x2 105 + -11x + -0.45x2 = 0.45x2 + -0.45x2 Combine like terms: 0.45x2 + -0.45x2 = 0.00 105 + -11x + -0.45x2 = 0.00 Begin completing the square. Divide all terms by -0.45 the coefficient of the squared term: Divide each side by '-0.45'. -233.3333333 + 24.44444444x + x2 = 0 Move the constant term to the right: Add '233.3333333' to each side of the equation. -233.3333333 + 24.44444444x + 233.3333333 + x2 = 0 + 233.3333333 Reorder the terms: -233.3333333 + 233.3333333 + 24.44444444x + x2 = 0 + 233.3333333 Combine like terms: -233.3333333 + 233.3333333 = 0.0000000 0.0000000 + 24.44444444x + x2 = 0 + 233.3333333 24.44444444x + x2 = 0 + 233.3333333 Combine like terms: 0 + 233.3333333 = 233.3333333 24.44444444x + x2 = 233.3333333 The x term is 24.44444444x. Take half its coefficient (12.22222222). Square it (149.3827160) and add it to both sides. Add '149.3827160' to each side of the equation. 24.44444444x + 149.3827160 + x2 = 233.3333333 + 149.3827160 Reorder the terms: 149.3827160 + 24.44444444x + x2 = 233.3333333 + 149.3827160 Combine like terms: 233.3333333 + 149.3827160 = 382.7160493 149.3827160 + 24.44444444x + x2 = 382.7160493 Factor a perfect square on the left side: (x + 12.22222222)(x + 12.22222222) = 382.7160493 Calculate the square root of the right side: 19.563129844 Break this problem into two subproblems by setting (x + 12.22222222) equal to 19.563129844 and -19.563129844.Subproblem 1
x + 12.22222222 = 19.563129844 Simplifying x + 12.22222222 = 19.563129844 Reorder the terms: 12.22222222 + x = 19.563129844 Solving 12.22222222 + x = 19.563129844 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.22222222' to each side of the equation. 12.22222222 + -12.22222222 + x = 19.563129844 + -12.22222222 Combine like terms: 12.22222222 + -12.22222222 = 0.00000000 0.00000000 + x = 19.563129844 + -12.22222222 x = 19.563129844 + -12.22222222 Combine like terms: 19.563129844 + -12.22222222 = 7.340907624 x = 7.340907624 Simplifying x = 7.340907624Subproblem 2
x + 12.22222222 = -19.563129844 Simplifying x + 12.22222222 = -19.563129844 Reorder the terms: 12.22222222 + x = -19.563129844 Solving 12.22222222 + x = -19.563129844 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.22222222' to each side of the equation. 12.22222222 + -12.22222222 + x = -19.563129844 + -12.22222222 Combine like terms: 12.22222222 + -12.22222222 = 0.00000000 0.00000000 + x = -19.563129844 + -12.22222222 x = -19.563129844 + -12.22222222 Combine like terms: -19.563129844 + -12.22222222 = -31.785352064 x = -31.785352064 Simplifying x = -31.785352064Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.340907624, -31.785352064}
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