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Simplifying 3x(5 + -2x) + 24 = 1 (5 * 3x + -2x * 3x) + 24 = 1 (15x + -6x2) + 24 = 1 Reorder the terms: 24 + 15x + -6x2 = 1 Solving 24 + 15x + -6x2 = 1 Solving for variable 'x'. Reorder the terms: 24 + -1 + 15x + -6x2 = 1 + -1 Combine like terms: 24 + -1 = 23 23 + 15x + -6x2 = 1 + -1 Combine like terms: 1 + -1 = 0 23 + 15x + -6x2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -3.833333333 + -2.5x + x2 = 0 Move the constant term to the right: Add '3.833333333' to each side of the equation. -3.833333333 + -2.5x + 3.833333333 + x2 = 0 + 3.833333333 Reorder the terms: -3.833333333 + 3.833333333 + -2.5x + x2 = 0 + 3.833333333 Combine like terms: -3.833333333 + 3.833333333 = 0.000000000 0.000000000 + -2.5x + x2 = 0 + 3.833333333 -2.5x + x2 = 0 + 3.833333333 Combine like terms: 0 + 3.833333333 = 3.833333333 -2.5x + x2 = 3.833333333 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = 3.833333333 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = 3.833333333 + 1.5625 Combine like terms: 3.833333333 + 1.5625 = 5.395833333 1.5625 + -2.5x + x2 = 5.395833333 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 5.395833333 Calculate the square root of the right side: 2.322893311 Break this problem into two subproblems by setting (x + -1.25) equal to 2.322893311 and -2.322893311.Subproblem 1
x + -1.25 = 2.322893311 Simplifying x + -1.25 = 2.322893311 Reorder the terms: -1.25 + x = 2.322893311 Solving -1.25 + x = 2.322893311 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 2.322893311 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 2.322893311 + 1.25 x = 2.322893311 + 1.25 Combine like terms: 2.322893311 + 1.25 = 3.572893311 x = 3.572893311 Simplifying x = 3.572893311Subproblem 2
x + -1.25 = -2.322893311 Simplifying x + -1.25 = -2.322893311 Reorder the terms: -1.25 + x = -2.322893311 Solving -1.25 + x = -2.322893311 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -2.322893311 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -2.322893311 + 1.25 x = -2.322893311 + 1.25 Combine like terms: -2.322893311 + 1.25 = -1.072893311 x = -1.072893311 Simplifying x = -1.072893311Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.572893311, -1.072893311}
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