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Simplifying 7x2 + 15x + -52 = -16x + 19 Reorder the terms: -52 + 15x + 7x2 = -16x + 19 Reorder the terms: -52 + 15x + 7x2 = 19 + -16x Solving -52 + 15x + 7x2 = 19 + -16x Solving for variable 'x'. Reorder the terms: -52 + -19 + 15x + 16x + 7x2 = 19 + -16x + -19 + 16x Combine like terms: -52 + -19 = -71 -71 + 15x + 16x + 7x2 = 19 + -16x + -19 + 16x Combine like terms: 15x + 16x = 31x -71 + 31x + 7x2 = 19 + -16x + -19 + 16x Reorder the terms: -71 + 31x + 7x2 = 19 + -19 + -16x + 16x Combine like terms: 19 + -19 = 0 -71 + 31x + 7x2 = 0 + -16x + 16x -71 + 31x + 7x2 = -16x + 16x Combine like terms: -16x + 16x = 0 -71 + 31x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -10.14285714 + 4.428571429x + x2 = 0 Move the constant term to the right: Add '10.14285714' to each side of the equation. -10.14285714 + 4.428571429x + 10.14285714 + x2 = 0 + 10.14285714 Reorder the terms: -10.14285714 + 10.14285714 + 4.428571429x + x2 = 0 + 10.14285714 Combine like terms: -10.14285714 + 10.14285714 = 0.00000000 0.00000000 + 4.428571429x + x2 = 0 + 10.14285714 4.428571429x + x2 = 0 + 10.14285714 Combine like terms: 0 + 10.14285714 = 10.14285714 4.428571429x + x2 = 10.14285714 The x term is 4.428571429x. Take half its coefficient (2.214285715). Square it (4.903061228) and add it to both sides. Add '4.903061228' to each side of the equation. 4.428571429x + 4.903061228 + x2 = 10.14285714 + 4.903061228 Reorder the terms: 4.903061228 + 4.428571429x + x2 = 10.14285714 + 4.903061228 Combine like terms: 10.14285714 + 4.903061228 = 15.045918368 4.903061228 + 4.428571429x + x2 = 15.045918368 Factor a perfect square on the left side: (x + 2.214285715)(x + 2.214285715) = 15.045918368 Calculate the square root of the right side: 3.878906852 Break this problem into two subproblems by setting (x + 2.214285715) equal to 3.878906852 and -3.878906852.Subproblem 1
x + 2.214285715 = 3.878906852 Simplifying x + 2.214285715 = 3.878906852 Reorder the terms: 2.214285715 + x = 3.878906852 Solving 2.214285715 + x = 3.878906852 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.214285715' to each side of the equation. 2.214285715 + -2.214285715 + x = 3.878906852 + -2.214285715 Combine like terms: 2.214285715 + -2.214285715 = 0.000000000 0.000000000 + x = 3.878906852 + -2.214285715 x = 3.878906852 + -2.214285715 Combine like terms: 3.878906852 + -2.214285715 = 1.664621137 x = 1.664621137 Simplifying x = 1.664621137Subproblem 2
x + 2.214285715 = -3.878906852 Simplifying x + 2.214285715 = -3.878906852 Reorder the terms: 2.214285715 + x = -3.878906852 Solving 2.214285715 + x = -3.878906852 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.214285715' to each side of the equation. 2.214285715 + -2.214285715 + x = -3.878906852 + -2.214285715 Combine like terms: 2.214285715 + -2.214285715 = 0.000000000 0.000000000 + x = -3.878906852 + -2.214285715 x = -3.878906852 + -2.214285715 Combine like terms: -3.878906852 + -2.214285715 = -6.093192567 x = -6.093192567 Simplifying x = -6.093192567Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.664621137, -6.093192567}
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