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Simplifying 5x + 76 = 9x2 + 24x + 36 Reorder the terms: 76 + 5x = 9x2 + 24x + 36 Reorder the terms: 76 + 5x = 36 + 24x + 9x2 Solving 76 + 5x = 36 + 24x + 9x2 Solving for variable 'x'. Reorder the terms: 76 + -36 + 5x + -24x + -9x2 = 36 + 24x + 9x2 + -36 + -24x + -9x2 Combine like terms: 76 + -36 = 40 40 + 5x + -24x + -9x2 = 36 + 24x + 9x2 + -36 + -24x + -9x2 Combine like terms: 5x + -24x = -19x 40 + -19x + -9x2 = 36 + 24x + 9x2 + -36 + -24x + -9x2 Reorder the terms: 40 + -19x + -9x2 = 36 + -36 + 24x + -24x + 9x2 + -9x2 Combine like terms: 36 + -36 = 0 40 + -19x + -9x2 = 0 + 24x + -24x + 9x2 + -9x2 40 + -19x + -9x2 = 24x + -24x + 9x2 + -9x2 Combine like terms: 24x + -24x = 0 40 + -19x + -9x2 = 0 + 9x2 + -9x2 40 + -19x + -9x2 = 9x2 + -9x2 Combine like terms: 9x2 + -9x2 = 0 40 + -19x + -9x2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -4.444444444 + 2.111111111x + x2 = 0 Move the constant term to the right: Add '4.444444444' to each side of the equation. -4.444444444 + 2.111111111x + 4.444444444 + x2 = 0 + 4.444444444 Reorder the terms: -4.444444444 + 4.444444444 + 2.111111111x + x2 = 0 + 4.444444444 Combine like terms: -4.444444444 + 4.444444444 = 0.000000000 0.000000000 + 2.111111111x + x2 = 0 + 4.444444444 2.111111111x + x2 = 0 + 4.444444444 Combine like terms: 0 + 4.444444444 = 4.444444444 2.111111111x + x2 = 4.444444444 The x term is 2.111111111x. Take half its coefficient (1.055555556). Square it (1.114197532) and add it to both sides. Add '1.114197532' to each side of the equation. 2.111111111x + 1.114197532 + x2 = 4.444444444 + 1.114197532 Reorder the terms: 1.114197532 + 2.111111111x + x2 = 4.444444444 + 1.114197532 Combine like terms: 4.444444444 + 1.114197532 = 5.558641976 1.114197532 + 2.111111111x + x2 = 5.558641976 Factor a perfect square on the left side: (x + 1.055555556)(x + 1.055555556) = 5.558641976 Calculate the square root of the right side: 2.357677242 Break this problem into two subproblems by setting (x + 1.055555556) equal to 2.357677242 and -2.357677242.Subproblem 1
x + 1.055555556 = 2.357677242 Simplifying x + 1.055555556 = 2.357677242 Reorder the terms: 1.055555556 + x = 2.357677242 Solving 1.055555556 + x = 2.357677242 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.055555556' to each side of the equation. 1.055555556 + -1.055555556 + x = 2.357677242 + -1.055555556 Combine like terms: 1.055555556 + -1.055555556 = 0.000000000 0.000000000 + x = 2.357677242 + -1.055555556 x = 2.357677242 + -1.055555556 Combine like terms: 2.357677242 + -1.055555556 = 1.302121686 x = 1.302121686 Simplifying x = 1.302121686Subproblem 2
x + 1.055555556 = -2.357677242 Simplifying x + 1.055555556 = -2.357677242 Reorder the terms: 1.055555556 + x = -2.357677242 Solving 1.055555556 + x = -2.357677242 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.055555556' to each side of the equation. 1.055555556 + -1.055555556 + x = -2.357677242 + -1.055555556 Combine like terms: 1.055555556 + -1.055555556 = 0.000000000 0.000000000 + x = -2.357677242 + -1.055555556 x = -2.357677242 + -1.055555556 Combine like terms: -2.357677242 + -1.055555556 = -3.413232798 x = -3.413232798 Simplifying x = -3.413232798Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.302121686, -3.413232798}
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