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Simplifying 4x2 + 31x + -19 = 2 Reorder the terms: -19 + 31x + 4x2 = 2 Solving -19 + 31x + 4x2 = 2 Solving for variable 'x'. Reorder the terms: -19 + -2 + 31x + 4x2 = 2 + -2 Combine like terms: -19 + -2 = -21 -21 + 31x + 4x2 = 2 + -2 Combine like terms: 2 + -2 = 0 -21 + 31x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5.25 + 7.75x + x2 = 0 Move the constant term to the right: Add '5.25' to each side of the equation. -5.25 + 7.75x + 5.25 + x2 = 0 + 5.25 Reorder the terms: -5.25 + 5.25 + 7.75x + x2 = 0 + 5.25 Combine like terms: -5.25 + 5.25 = 0.00 0.00 + 7.75x + x2 = 0 + 5.25 7.75x + x2 = 0 + 5.25 Combine like terms: 0 + 5.25 = 5.25 7.75x + x2 = 5.25 The x term is 7.75x. Take half its coefficient (3.875). Square it (15.015625) and add it to both sides. Add '15.015625' to each side of the equation. 7.75x + 15.015625 + x2 = 5.25 + 15.015625 Reorder the terms: 15.015625 + 7.75x + x2 = 5.25 + 15.015625 Combine like terms: 5.25 + 15.015625 = 20.265625 15.015625 + 7.75x + x2 = 20.265625 Factor a perfect square on the left side: (x + 3.875)(x + 3.875) = 20.265625 Calculate the square root of the right side: 4.501735776 Break this problem into two subproblems by setting (x + 3.875) equal to 4.501735776 and -4.501735776.Subproblem 1
x + 3.875 = 4.501735776 Simplifying x + 3.875 = 4.501735776 Reorder the terms: 3.875 + x = 4.501735776 Solving 3.875 + x = 4.501735776 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.875' to each side of the equation. 3.875 + -3.875 + x = 4.501735776 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + x = 4.501735776 + -3.875 x = 4.501735776 + -3.875 Combine like terms: 4.501735776 + -3.875 = 0.626735776 x = 0.626735776 Simplifying x = 0.626735776Subproblem 2
x + 3.875 = -4.501735776 Simplifying x + 3.875 = -4.501735776 Reorder the terms: 3.875 + x = -4.501735776 Solving 3.875 + x = -4.501735776 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.875' to each side of the equation. 3.875 + -3.875 + x = -4.501735776 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + x = -4.501735776 + -3.875 x = -4.501735776 + -3.875 Combine like terms: -4.501735776 + -3.875 = -8.376735776 x = -8.376735776 Simplifying x = -8.376735776Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.626735776, -8.376735776}
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