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Simplifying 52x = 7x2 + -67 Reorder the terms: 52x = -67 + 7x2 Solving 52x = -67 + 7x2 Solving for variable 'x'. Reorder the terms: 67 + 52x + -7x2 = -67 + 7x2 + 67 + -7x2 Reorder the terms: 67 + 52x + -7x2 = -67 + 67 + 7x2 + -7x2 Combine like terms: -67 + 67 = 0 67 + 52x + -7x2 = 0 + 7x2 + -7x2 67 + 52x + -7x2 = 7x2 + -7x2 Combine like terms: 7x2 + -7x2 = 0 67 + 52x + -7x2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -9.571428571 + -7.428571429x + x2 = 0 Move the constant term to the right: Add '9.571428571' to each side of the equation. -9.571428571 + -7.428571429x + 9.571428571 + x2 = 0 + 9.571428571 Reorder the terms: -9.571428571 + 9.571428571 + -7.428571429x + x2 = 0 + 9.571428571 Combine like terms: -9.571428571 + 9.571428571 = 0.000000000 0.000000000 + -7.428571429x + x2 = 0 + 9.571428571 -7.428571429x + x2 = 0 + 9.571428571 Combine like terms: 0 + 9.571428571 = 9.571428571 -7.428571429x + x2 = 9.571428571 The x term is -7.428571429x. Take half its coefficient (-3.714285715). Square it (13.79591837) and add it to both sides. Add '13.79591837' to each side of the equation. -7.428571429x + 13.79591837 + x2 = 9.571428571 + 13.79591837 Reorder the terms: 13.79591837 + -7.428571429x + x2 = 9.571428571 + 13.79591837 Combine like terms: 9.571428571 + 13.79591837 = 23.367346941 13.79591837 + -7.428571429x + x2 = 23.367346941 Factor a perfect square on the left side: (x + -3.714285715)(x + -3.714285715) = 23.367346941 Calculate the square root of the right side: 4.833978376 Break this problem into two subproblems by setting (x + -3.714285715) equal to 4.833978376 and -4.833978376.Subproblem 1
x + -3.714285715 = 4.833978376 Simplifying x + -3.714285715 = 4.833978376 Reorder the terms: -3.714285715 + x = 4.833978376 Solving -3.714285715 + x = 4.833978376 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.714285715' to each side of the equation. -3.714285715 + 3.714285715 + x = 4.833978376 + 3.714285715 Combine like terms: -3.714285715 + 3.714285715 = 0.000000000 0.000000000 + x = 4.833978376 + 3.714285715 x = 4.833978376 + 3.714285715 Combine like terms: 4.833978376 + 3.714285715 = 8.548264091 x = 8.548264091 Simplifying x = 8.548264091Subproblem 2
x + -3.714285715 = -4.833978376 Simplifying x + -3.714285715 = -4.833978376 Reorder the terms: -3.714285715 + x = -4.833978376 Solving -3.714285715 + x = -4.833978376 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.714285715' to each side of the equation. -3.714285715 + 3.714285715 + x = -4.833978376 + 3.714285715 Combine like terms: -3.714285715 + 3.714285715 = 0.000000000 0.000000000 + x = -4.833978376 + 3.714285715 x = -4.833978376 + 3.714285715 Combine like terms: -4.833978376 + 3.714285715 = -1.119692661 x = -1.119692661 Simplifying x = -1.119692661Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.548264091, -1.119692661}
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