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Simplifying (6x2 + -6x + -74) + (7x2 + 9x) = 0 Reorder the terms: (-74 + -6x + 6x2) + (7x2 + 9x) = 0 Remove parenthesis around (-74 + -6x + 6x2) -74 + -6x + 6x2 + (7x2 + 9x) = 0 Reorder the terms: -74 + -6x + 6x2 + (9x + 7x2) = 0 Remove parenthesis around (9x + 7x2) -74 + -6x + 6x2 + 9x + 7x2 = 0 Reorder the terms: -74 + -6x + 9x + 6x2 + 7x2 = 0 Combine like terms: -6x + 9x = 3x -74 + 3x + 6x2 + 7x2 = 0 Combine like terms: 6x2 + 7x2 = 13x2 -74 + 3x + 13x2 = 0 Solving -74 + 3x + 13x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -5.692307692 + 0.2307692308x + x2 = 0 Move the constant term to the right: Add '5.692307692' to each side of the equation. -5.692307692 + 0.2307692308x + 5.692307692 + x2 = 0 + 5.692307692 Reorder the terms: -5.692307692 + 5.692307692 + 0.2307692308x + x2 = 0 + 5.692307692 Combine like terms: -5.692307692 + 5.692307692 = 0.000000000 0.000000000 + 0.2307692308x + x2 = 0 + 5.692307692 0.2307692308x + x2 = 0 + 5.692307692 Combine like terms: 0 + 5.692307692 = 5.692307692 0.2307692308x + x2 = 5.692307692 The x term is 0.2307692308x. Take half its coefficient (0.1153846154). Square it (0.01331360947) and add it to both sides. Add '0.01331360947' to each side of the equation. 0.2307692308x + 0.01331360947 + x2 = 5.692307692 + 0.01331360947 Reorder the terms: 0.01331360947 + 0.2307692308x + x2 = 5.692307692 + 0.01331360947 Combine like terms: 5.692307692 + 0.01331360947 = 5.70562130147 0.01331360947 + 0.2307692308x + x2 = 5.70562130147 Factor a perfect square on the left side: (x + 0.1153846154)(x + 0.1153846154) = 5.70562130147 Calculate the square root of the right side: 2.388644239 Break this problem into two subproblems by setting (x + 0.1153846154) equal to 2.388644239 and -2.388644239.Subproblem 1
x + 0.1153846154 = 2.388644239 Simplifying x + 0.1153846154 = 2.388644239 Reorder the terms: 0.1153846154 + x = 2.388644239 Solving 0.1153846154 + x = 2.388644239 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1153846154' to each side of the equation. 0.1153846154 + -0.1153846154 + x = 2.388644239 + -0.1153846154 Combine like terms: 0.1153846154 + -0.1153846154 = 0.0000000000 0.0000000000 + x = 2.388644239 + -0.1153846154 x = 2.388644239 + -0.1153846154 Combine like terms: 2.388644239 + -0.1153846154 = 2.2732596236 x = 2.2732596236 Simplifying x = 2.2732596236Subproblem 2
x + 0.1153846154 = -2.388644239 Simplifying x + 0.1153846154 = -2.388644239 Reorder the terms: 0.1153846154 + x = -2.388644239 Solving 0.1153846154 + x = -2.388644239 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1153846154' to each side of the equation. 0.1153846154 + -0.1153846154 + x = -2.388644239 + -0.1153846154 Combine like terms: 0.1153846154 + -0.1153846154 = 0.0000000000 0.0000000000 + x = -2.388644239 + -0.1153846154 x = -2.388644239 + -0.1153846154 Combine like terms: -2.388644239 + -0.1153846154 = -2.5040288544 x = -2.5040288544 Simplifying x = -2.5040288544Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.2732596236, -2.5040288544}
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