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Simplifying 5x + 76 = 9x2 + 54x + 36 Reorder the terms: 76 + 5x = 9x2 + 54x + 36 Reorder the terms: 76 + 5x = 36 + 54x + 9x2 Solving 76 + 5x = 36 + 54x + 9x2 Solving for variable 'x'. Reorder the terms: 76 + -36 + 5x + -54x + -9x2 = 36 + 54x + 9x2 + -36 + -54x + -9x2 Combine like terms: 76 + -36 = 40 40 + 5x + -54x + -9x2 = 36 + 54x + 9x2 + -36 + -54x + -9x2 Combine like terms: 5x + -54x = -49x 40 + -49x + -9x2 = 36 + 54x + 9x2 + -36 + -54x + -9x2 Reorder the terms: 40 + -49x + -9x2 = 36 + -36 + 54x + -54x + 9x2 + -9x2 Combine like terms: 36 + -36 = 0 40 + -49x + -9x2 = 0 + 54x + -54x + 9x2 + -9x2 40 + -49x + -9x2 = 54x + -54x + 9x2 + -9x2 Combine like terms: 54x + -54x = 0 40 + -49x + -9x2 = 0 + 9x2 + -9x2 40 + -49x + -9x2 = 9x2 + -9x2 Combine like terms: 9x2 + -9x2 = 0 40 + -49x + -9x2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -4.444444444 + 5.444444444x + x2 = 0 Move the constant term to the right: Add '4.444444444' to each side of the equation. -4.444444444 + 5.444444444x + 4.444444444 + x2 = 0 + 4.444444444 Reorder the terms: -4.444444444 + 4.444444444 + 5.444444444x + x2 = 0 + 4.444444444 Combine like terms: -4.444444444 + 4.444444444 = 0.000000000 0.000000000 + 5.444444444x + x2 = 0 + 4.444444444 5.444444444x + x2 = 0 + 4.444444444 Combine like terms: 0 + 4.444444444 = 4.444444444 5.444444444x + x2 = 4.444444444 The x term is 5.444444444x. Take half its coefficient (2.722222222). Square it (7.410493826) and add it to both sides. Add '7.410493826' to each side of the equation. 5.444444444x + 7.410493826 + x2 = 4.444444444 + 7.410493826 Reorder the terms: 7.410493826 + 5.444444444x + x2 = 4.444444444 + 7.410493826 Combine like terms: 4.444444444 + 7.410493826 = 11.85493827 7.410493826 + 5.444444444x + x2 = 11.85493827 Factor a perfect square on the left side: (x + 2.722222222)(x + 2.722222222) = 11.85493827 Calculate the square root of the right side: 3.443100096 Break this problem into two subproblems by setting (x + 2.722222222) equal to 3.443100096 and -3.443100096.Subproblem 1
x + 2.722222222 = 3.443100096 Simplifying x + 2.722222222 = 3.443100096 Reorder the terms: 2.722222222 + x = 3.443100096 Solving 2.722222222 + x = 3.443100096 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.722222222' to each side of the equation. 2.722222222 + -2.722222222 + x = 3.443100096 + -2.722222222 Combine like terms: 2.722222222 + -2.722222222 = 0.000000000 0.000000000 + x = 3.443100096 + -2.722222222 x = 3.443100096 + -2.722222222 Combine like terms: 3.443100096 + -2.722222222 = 0.720877874 x = 0.720877874 Simplifying x = 0.720877874Subproblem 2
x + 2.722222222 = -3.443100096 Simplifying x + 2.722222222 = -3.443100096 Reorder the terms: 2.722222222 + x = -3.443100096 Solving 2.722222222 + x = -3.443100096 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.722222222' to each side of the equation. 2.722222222 + -2.722222222 + x = -3.443100096 + -2.722222222 Combine like terms: 2.722222222 + -2.722222222 = 0.000000000 0.000000000 + x = -3.443100096 + -2.722222222 x = -3.443100096 + -2.722222222 Combine like terms: -3.443100096 + -2.722222222 = -6.165322318 x = -6.165322318 Simplifying x = -6.165322318Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.720877874, -6.165322318}
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