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Simplifying 3x2 + 11x + -75 = 0 Reorder the terms: -75 + 11x + 3x2 = 0 Solving -75 + 11x + 3x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -25 + 3.666666667x + x2 = 0 Move the constant term to the right: Add '25' to each side of the equation. -25 + 3.666666667x + 25 + x2 = 0 + 25 Reorder the terms: -25 + 25 + 3.666666667x + x2 = 0 + 25 Combine like terms: -25 + 25 = 0 0 + 3.666666667x + x2 = 0 + 25 3.666666667x + x2 = 0 + 25 Combine like terms: 0 + 25 = 25 3.666666667x + x2 = 25 The x term is 3.666666667x. Take half its coefficient (1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. 3.666666667x + 3.361111114 + x2 = 25 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667x + x2 = 25 + 3.361111114 Combine like terms: 25 + 3.361111114 = 28.361111114 3.361111114 + 3.666666667x + x2 = 28.361111114 Factor a perfect square on the left side: (x + 1.833333334)(x + 1.833333334) = 28.361111114 Calculate the square root of the right side: 5.325515103 Break this problem into two subproblems by setting (x + 1.833333334) equal to 5.325515103 and -5.325515103.Subproblem 1
x + 1.833333334 = 5.325515103 Simplifying x + 1.833333334 = 5.325515103 Reorder the terms: 1.833333334 + x = 5.325515103 Solving 1.833333334 + x = 5.325515103 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = 5.325515103 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = 5.325515103 + -1.833333334 x = 5.325515103 + -1.833333334 Combine like terms: 5.325515103 + -1.833333334 = 3.492181769 x = 3.492181769 Simplifying x = 3.492181769Subproblem 2
x + 1.833333334 = -5.325515103 Simplifying x + 1.833333334 = -5.325515103 Reorder the terms: 1.833333334 + x = -5.325515103 Solving 1.833333334 + x = -5.325515103 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = -5.325515103 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = -5.325515103 + -1.833333334 x = -5.325515103 + -1.833333334 Combine like terms: -5.325515103 + -1.833333334 = -7.158848437 x = -7.158848437 Simplifying x = -7.158848437Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.492181769, -7.158848437}
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