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Simplifying 2.5(3 + x) * 2.5x + 7.5 = 0 Reorder the terms for easier multiplication: 2.5 * 2.5x(3 + x) + 7.5 = 0 Multiply 2.5 * 2.5 6.25x(3 + x) + 7.5 = 0 (3 * 6.25x + x * 6.25x) + 7.5 = 0 (18.75x + 6.25x2) + 7.5 = 0 Reorder the terms: 7.5 + 18.75x + 6.25x2 = 0 Solving 7.5 + 18.75x + 6.25x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 6.25 the coefficient of the squared term: Divide each side by '6.25'. 1.2 + 3x + x2 = 0 Move the constant term to the right: Add '-1.2' to each side of the equation. 1.2 + 3x + -1.2 + x2 = 0 + -1.2 Reorder the terms: 1.2 + -1.2 + 3x + x2 = 0 + -1.2 Combine like terms: 1.2 + -1.2 = 0.0 0.0 + 3x + x2 = 0 + -1.2 3x + x2 = 0 + -1.2 Combine like terms: 0 + -1.2 = -1.2 3x + x2 = -1.2 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = -1.2 + 2.25 Reorder the terms: 2.25 + 3x + x2 = -1.2 + 2.25 Combine like terms: -1.2 + 2.25 = 1.05 2.25 + 3x + x2 = 1.05 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 1.05 Calculate the square root of the right side: 1.024695077 Break this problem into two subproblems by setting (x + 1.5) equal to 1.024695077 and -1.024695077.Subproblem 1
x + 1.5 = 1.024695077 Simplifying x + 1.5 = 1.024695077 Reorder the terms: 1.5 + x = 1.024695077 Solving 1.5 + x = 1.024695077 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 1.024695077 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.024695077 + -1.5 x = 1.024695077 + -1.5 Combine like terms: 1.024695077 + -1.5 = -0.475304923 x = -0.475304923 Simplifying x = -0.475304923Subproblem 2
x + 1.5 = -1.024695077 Simplifying x + 1.5 = -1.024695077 Reorder the terms: 1.5 + x = -1.024695077 Solving 1.5 + x = -1.024695077 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -1.024695077 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.024695077 + -1.5 x = -1.024695077 + -1.5 Combine like terms: -1.024695077 + -1.5 = -2.524695077 x = -2.524695077 Simplifying x = -2.524695077Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.475304923, -2.524695077}
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