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Simplifying 12x2 + 37x + 6 = 67 Reorder the terms: 6 + 37x + 12x2 = 67 Solving 6 + 37x + 12x2 = 67 Solving for variable 'x'. Reorder the terms: 6 + -67 + 37x + 12x2 = 67 + -67 Combine like terms: 6 + -67 = -61 -61 + 37x + 12x2 = 67 + -67 Combine like terms: 67 + -67 = 0 -61 + 37x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -5.083333333 + 3.083333333x + x2 = 0 Move the constant term to the right: Add '5.083333333' to each side of the equation. -5.083333333 + 3.083333333x + 5.083333333 + x2 = 0 + 5.083333333 Reorder the terms: -5.083333333 + 5.083333333 + 3.083333333x + x2 = 0 + 5.083333333 Combine like terms: -5.083333333 + 5.083333333 = 0.000000000 0.000000000 + 3.083333333x + x2 = 0 + 5.083333333 3.083333333x + x2 = 0 + 5.083333333 Combine like terms: 0 + 5.083333333 = 5.083333333 3.083333333x + x2 = 5.083333333 The x term is 3.083333333x. Take half its coefficient (1.541666667). Square it (2.376736112) and add it to both sides. Add '2.376736112' to each side of the equation. 3.083333333x + 2.376736112 + x2 = 5.083333333 + 2.376736112 Reorder the terms: 2.376736112 + 3.083333333x + x2 = 5.083333333 + 2.376736112 Combine like terms: 5.083333333 + 2.376736112 = 7.460069445 2.376736112 + 3.083333333x + x2 = 7.460069445 Factor a perfect square on the left side: (x + 1.541666667)(x + 1.541666667) = 7.460069445 Calculate the square root of the right side: 2.73131277 Break this problem into two subproblems by setting (x + 1.541666667) equal to 2.73131277 and -2.73131277.Subproblem 1
x + 1.541666667 = 2.73131277 Simplifying x + 1.541666667 = 2.73131277 Reorder the terms: 1.541666667 + x = 2.73131277 Solving 1.541666667 + x = 2.73131277 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.541666667' to each side of the equation. 1.541666667 + -1.541666667 + x = 2.73131277 + -1.541666667 Combine like terms: 1.541666667 + -1.541666667 = 0.000000000 0.000000000 + x = 2.73131277 + -1.541666667 x = 2.73131277 + -1.541666667 Combine like terms: 2.73131277 + -1.541666667 = 1.189646103 x = 1.189646103 Simplifying x = 1.189646103Subproblem 2
x + 1.541666667 = -2.73131277 Simplifying x + 1.541666667 = -2.73131277 Reorder the terms: 1.541666667 + x = -2.73131277 Solving 1.541666667 + x = -2.73131277 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.541666667' to each side of the equation. 1.541666667 + -1.541666667 + x = -2.73131277 + -1.541666667 Combine like terms: 1.541666667 + -1.541666667 = 0.000000000 0.000000000 + x = -2.73131277 + -1.541666667 x = -2.73131277 + -1.541666667 Combine like terms: -2.73131277 + -1.541666667 = -4.272979437 x = -4.272979437 Simplifying x = -4.272979437Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.189646103, -4.272979437}
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