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Simplifying 0.5x(x + -82) = 19 + 8x Reorder the terms: 0.5x(-82 + x) = 19 + 8x (-82 * 0.5x + x * 0.5x) = 19 + 8x (-41x + 0.5x2) = 19 + 8x Solving -41x + 0.5x2 = 19 + 8x Solving for variable 'x'. Reorder the terms: -19 + -41x + -8x + 0.5x2 = 19 + 8x + -19 + -8x Combine like terms: -41x + -8x = -49x -19 + -49x + 0.5x2 = 19 + 8x + -19 + -8x Reorder the terms: -19 + -49x + 0.5x2 = 19 + -19 + 8x + -8x Combine like terms: 19 + -19 = 0 -19 + -49x + 0.5x2 = 0 + 8x + -8x -19 + -49x + 0.5x2 = 8x + -8x Combine like terms: 8x + -8x = 0 -19 + -49x + 0.5x2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -38 + -98x + x2 = 0 Move the constant term to the right: Add '38' to each side of the equation. -38 + -98x + 38 + x2 = 0 + 38 Reorder the terms: -38 + 38 + -98x + x2 = 0 + 38 Combine like terms: -38 + 38 = 0 0 + -98x + x2 = 0 + 38 -98x + x2 = 0 + 38 Combine like terms: 0 + 38 = 38 -98x + x2 = 38 The x term is -98x. Take half its coefficient (-49). Square it (2401) and add it to both sides. Add '2401' to each side of the equation. -98x + 2401 + x2 = 38 + 2401 Reorder the terms: 2401 + -98x + x2 = 38 + 2401 Combine like terms: 38 + 2401 = 2439 2401 + -98x + x2 = 2439 Factor a perfect square on the left side: (x + -49)(x + -49) = 2439 Calculate the square root of the right side: 49.386232899 Break this problem into two subproblems by setting (x + -49) equal to 49.386232899 and -49.386232899.Subproblem 1
x + -49 = 49.386232899 Simplifying x + -49 = 49.386232899 Reorder the terms: -49 + x = 49.386232899 Solving -49 + x = 49.386232899 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '49' to each side of the equation. -49 + 49 + x = 49.386232899 + 49 Combine like terms: -49 + 49 = 0 0 + x = 49.386232899 + 49 x = 49.386232899 + 49 Combine like terms: 49.386232899 + 49 = 98.386232899 x = 98.386232899 Simplifying x = 98.386232899Subproblem 2
x + -49 = -49.386232899 Simplifying x + -49 = -49.386232899 Reorder the terms: -49 + x = -49.386232899 Solving -49 + x = -49.386232899 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '49' to each side of the equation. -49 + 49 + x = -49.386232899 + 49 Combine like terms: -49 + 49 = 0 0 + x = -49.386232899 + 49 x = -49.386232899 + 49 Combine like terms: -49.386232899 + 49 = -0.386232899 x = -0.386232899 Simplifying x = -0.386232899Solution
The solution to the problem is based on the solutions from the subproblems. x = {98.386232899, -0.386232899}
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