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Simplifying -24x2 + 57x + 16 = 0 Reorder the terms: 16 + 57x + -24x2 = 0 Solving 16 + 57x + -24x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -24 the coefficient of the squared term: Divide each side by '-24'. -0.6666666667 + -2.375x + x2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -2.375x + 0.6666666667 + x2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -2.375x + x2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -2.375x + x2 = 0 + 0.6666666667 -2.375x + x2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -2.375x + x2 = 0.6666666667 The x term is -2.375x. Take half its coefficient (-1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. -2.375x + 1.41015625 + x2 = 0.6666666667 + 1.41015625 Reorder the terms: 1.41015625 + -2.375x + x2 = 0.6666666667 + 1.41015625 Combine like terms: 0.6666666667 + 1.41015625 = 2.0768229167 1.41015625 + -2.375x + x2 = 2.0768229167 Factor a perfect square on the left side: (x + -1.1875)(x + -1.1875) = 2.0768229167 Calculate the square root of the right side: 1.441118634 Break this problem into two subproblems by setting (x + -1.1875) equal to 1.441118634 and -1.441118634.Subproblem 1
x + -1.1875 = 1.441118634 Simplifying x + -1.1875 = 1.441118634 Reorder the terms: -1.1875 + x = 1.441118634 Solving -1.1875 + x = 1.441118634 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + x = 1.441118634 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + x = 1.441118634 + 1.1875 x = 1.441118634 + 1.1875 Combine like terms: 1.441118634 + 1.1875 = 2.628618634 x = 2.628618634 Simplifying x = 2.628618634Subproblem 2
x + -1.1875 = -1.441118634 Simplifying x + -1.1875 = -1.441118634 Reorder the terms: -1.1875 + x = -1.441118634 Solving -1.1875 + x = -1.441118634 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + x = -1.441118634 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + x = -1.441118634 + 1.1875 x = -1.441118634 + 1.1875 Combine like terms: -1.441118634 + 1.1875 = -0.253618634 x = -0.253618634 Simplifying x = -0.253618634Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.628618634, -0.253618634}
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