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Simplifying 0 = -0.5x2 + 36x + -109 Reorder the terms: 0 = -109 + 36x + -0.5x2 Solving 0 = -109 + 36x + -0.5x2 Solving for variable 'x'. Combine like terms: 0 + 109 = 109 109 + -36x + 0.5x2 = -109 + 36x + -0.5x2 + 109 + -36x + 0.5x2 Reorder the terms: 109 + -36x + 0.5x2 = -109 + 109 + 36x + -36x + -0.5x2 + 0.5x2 Combine like terms: -109 + 109 = 0 109 + -36x + 0.5x2 = 0 + 36x + -36x + -0.5x2 + 0.5x2 109 + -36x + 0.5x2 = 36x + -36x + -0.5x2 + 0.5x2 Combine like terms: 36x + -36x = 0 109 + -36x + 0.5x2 = 0 + -0.5x2 + 0.5x2 109 + -36x + 0.5x2 = -0.5x2 + 0.5x2 Combine like terms: -0.5x2 + 0.5x2 = 0.0 109 + -36x + 0.5x2 = 0.0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. 218 + -72x + x2 = 0 Move the constant term to the right: Add '-218' to each side of the equation. 218 + -72x + -218 + x2 = 0 + -218 Reorder the terms: 218 + -218 + -72x + x2 = 0 + -218 Combine like terms: 218 + -218 = 0 0 + -72x + x2 = 0 + -218 -72x + x2 = 0 + -218 Combine like terms: 0 + -218 = -218 -72x + x2 = -218 The x term is -72x. Take half its coefficient (-36). Square it (1296) and add it to both sides. Add '1296' to each side of the equation. -72x + 1296 + x2 = -218 + 1296 Reorder the terms: 1296 + -72x + x2 = -218 + 1296 Combine like terms: -218 + 1296 = 1078 1296 + -72x + x2 = 1078 Factor a perfect square on the left side: (x + -36)(x + -36) = 1078 Calculate the square root of the right side: 32.832910319 Break this problem into two subproblems by setting (x + -36) equal to 32.832910319 and -32.832910319.Subproblem 1
x + -36 = 32.832910319 Simplifying x + -36 = 32.832910319 Reorder the terms: -36 + x = 32.832910319 Solving -36 + x = 32.832910319 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '36' to each side of the equation. -36 + 36 + x = 32.832910319 + 36 Combine like terms: -36 + 36 = 0 0 + x = 32.832910319 + 36 x = 32.832910319 + 36 Combine like terms: 32.832910319 + 36 = 68.832910319 x = 68.832910319 Simplifying x = 68.832910319Subproblem 2
x + -36 = -32.832910319 Simplifying x + -36 = -32.832910319 Reorder the terms: -36 + x = -32.832910319 Solving -36 + x = -32.832910319 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '36' to each side of the equation. -36 + 36 + x = -32.832910319 + 36 Combine like terms: -36 + 36 = 0 0 + x = -32.832910319 + 36 x = -32.832910319 + 36 Combine like terms: -32.832910319 + 36 = 3.167089681 x = 3.167089681 Simplifying x = 3.167089681Solution
The solution to the problem is based on the solutions from the subproblems. x = {68.832910319, 3.167089681}
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