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Simplifying 0 = 7x2 + -24x + 15 Reorder the terms: 0 = 15 + -24x + 7x2 Solving 0 = 15 + -24x + 7x2 Solving for variable 'x'. Combine like terms: 0 + -15 = -15 -15 + 24x + -7x2 = 15 + -24x + 7x2 + -15 + 24x + -7x2 Reorder the terms: -15 + 24x + -7x2 = 15 + -15 + -24x + 24x + 7x2 + -7x2 Combine like terms: 15 + -15 = 0 -15 + 24x + -7x2 = 0 + -24x + 24x + 7x2 + -7x2 -15 + 24x + -7x2 = -24x + 24x + 7x2 + -7x2 Combine like terms: -24x + 24x = 0 -15 + 24x + -7x2 = 0 + 7x2 + -7x2 -15 + 24x + -7x2 = 7x2 + -7x2 Combine like terms: 7x2 + -7x2 = 0 -15 + 24x + -7x2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. 2.142857143 + -3.428571429x + x2 = 0 Move the constant term to the right: Add '-2.142857143' to each side of the equation. 2.142857143 + -3.428571429x + -2.142857143 + x2 = 0 + -2.142857143 Reorder the terms: 2.142857143 + -2.142857143 + -3.428571429x + x2 = 0 + -2.142857143 Combine like terms: 2.142857143 + -2.142857143 = 0.000000000 0.000000000 + -3.428571429x + x2 = 0 + -2.142857143 -3.428571429x + x2 = 0 + -2.142857143 Combine like terms: 0 + -2.142857143 = -2.142857143 -3.428571429x + x2 = -2.142857143 The x term is -3.428571429x. Take half its coefficient (-1.714285715). Square it (2.938775513) and add it to both sides. Add '2.938775513' to each side of the equation. -3.428571429x + 2.938775513 + x2 = -2.142857143 + 2.938775513 Reorder the terms: 2.938775513 + -3.428571429x + x2 = -2.142857143 + 2.938775513 Combine like terms: -2.142857143 + 2.938775513 = 0.79591837 2.938775513 + -3.428571429x + x2 = 0.79591837 Factor a perfect square on the left side: (x + -1.714285715)(x + -1.714285715) = 0.79591837 Calculate the square root of the right side: 0.892142573 Break this problem into two subproblems by setting (x + -1.714285715) equal to 0.892142573 and -0.892142573.Subproblem 1
x + -1.714285715 = 0.892142573 Simplifying x + -1.714285715 = 0.892142573 Reorder the terms: -1.714285715 + x = 0.892142573 Solving -1.714285715 + x = 0.892142573 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.714285715' to each side of the equation. -1.714285715 + 1.714285715 + x = 0.892142573 + 1.714285715 Combine like terms: -1.714285715 + 1.714285715 = 0.000000000 0.000000000 + x = 0.892142573 + 1.714285715 x = 0.892142573 + 1.714285715 Combine like terms: 0.892142573 + 1.714285715 = 2.606428288 x = 2.606428288 Simplifying x = 2.606428288Subproblem 2
x + -1.714285715 = -0.892142573 Simplifying x + -1.714285715 = -0.892142573 Reorder the terms: -1.714285715 + x = -0.892142573 Solving -1.714285715 + x = -0.892142573 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.714285715' to each side of the equation. -1.714285715 + 1.714285715 + x = -0.892142573 + 1.714285715 Combine like terms: -1.714285715 + 1.714285715 = 0.000000000 0.000000000 + x = -0.892142573 + 1.714285715 x = -0.892142573 + 1.714285715 Combine like terms: -0.892142573 + 1.714285715 = 0.822143142 x = 0.822143142 Simplifying x = 0.822143142Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.606428288, 0.822143142}
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