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Simplifying 5x(6 + 3x) = 17 + -2x (6 * 5x + 3x * 5x) = 17 + -2x (30x + 15x2) = 17 + -2x Solving 30x + 15x2 = 17 + -2x Solving for variable 'x'. Reorder the terms: -17 + 30x + 2x + 15x2 = 17 + -2x + -17 + 2x Combine like terms: 30x + 2x = 32x -17 + 32x + 15x2 = 17 + -2x + -17 + 2x Reorder the terms: -17 + 32x + 15x2 = 17 + -17 + -2x + 2x Combine like terms: 17 + -17 = 0 -17 + 32x + 15x2 = 0 + -2x + 2x -17 + 32x + 15x2 = -2x + 2x Combine like terms: -2x + 2x = 0 -17 + 32x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -1.133333333 + 2.133333333x + x2 = 0 Move the constant term to the right: Add '1.133333333' to each side of the equation. -1.133333333 + 2.133333333x + 1.133333333 + x2 = 0 + 1.133333333 Reorder the terms: -1.133333333 + 1.133333333 + 2.133333333x + x2 = 0 + 1.133333333 Combine like terms: -1.133333333 + 1.133333333 = 0.000000000 0.000000000 + 2.133333333x + x2 = 0 + 1.133333333 2.133333333x + x2 = 0 + 1.133333333 Combine like terms: 0 + 1.133333333 = 1.133333333 2.133333333x + x2 = 1.133333333 The x term is 2.133333333x. Take half its coefficient (1.066666667). Square it (1.137777778) and add it to both sides. Add '1.137777778' to each side of the equation. 2.133333333x + 1.137777778 + x2 = 1.133333333 + 1.137777778 Reorder the terms: 1.137777778 + 2.133333333x + x2 = 1.133333333 + 1.137777778 Combine like terms: 1.133333333 + 1.137777778 = 2.271111111 1.137777778 + 2.133333333x + x2 = 2.271111111 Factor a perfect square on the left side: (x + 1.066666667)(x + 1.066666667) = 2.271111111 Calculate the square root of the right side: 1.507020607 Break this problem into two subproblems by setting (x + 1.066666667) equal to 1.507020607 and -1.507020607.Subproblem 1
x + 1.066666667 = 1.507020607 Simplifying x + 1.066666667 = 1.507020607 Reorder the terms: 1.066666667 + x = 1.507020607 Solving 1.066666667 + x = 1.507020607 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.066666667' to each side of the equation. 1.066666667 + -1.066666667 + x = 1.507020607 + -1.066666667 Combine like terms: 1.066666667 + -1.066666667 = 0.000000000 0.000000000 + x = 1.507020607 + -1.066666667 x = 1.507020607 + -1.066666667 Combine like terms: 1.507020607 + -1.066666667 = 0.44035394 x = 0.44035394 Simplifying x = 0.44035394Subproblem 2
x + 1.066666667 = -1.507020607 Simplifying x + 1.066666667 = -1.507020607 Reorder the terms: 1.066666667 + x = -1.507020607 Solving 1.066666667 + x = -1.507020607 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.066666667' to each side of the equation. 1.066666667 + -1.066666667 + x = -1.507020607 + -1.066666667 Combine like terms: 1.066666667 + -1.066666667 = 0.000000000 0.000000000 + x = -1.507020607 + -1.066666667 x = -1.507020607 + -1.066666667 Combine like terms: -1.507020607 + -1.066666667 = -2.573687274 x = -2.573687274 Simplifying x = -2.573687274Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.44035394, -2.573687274}
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