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Simplifying 0.5[(4x + 6) * x] = 30 Reorder the terms: 0.5[(6 + 4x) * x] = 30 Reorder the terms for easier multiplication: 0.5[x(6 + 4x)] = 30 0.5[(6 * x + 4x * x)] = 30 0.5[(6x + 4x2)] = 30 [6x * 0.5 + 4x2 * 0.5] = 30 [3x + 2x2] = 30 Solving 3x + 2x2 = 30 Solving for variable 'x'. Reorder the terms: -30 + 3x + 2x2 = 30 + -30 Combine like terms: 30 + -30 = 0 -30 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -15 + 1.5x + x2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + 1.5x + 15 + x2 = 0 + 15 Reorder the terms: -15 + 15 + 1.5x + x2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 1.5x + x2 = 0 + 15 1.5x + x2 = 0 + 15 Combine like terms: 0 + 15 = 15 1.5x + x2 = 15 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 15 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 15 + 0.5625 Combine like terms: 15 + 0.5625 = 15.5625 0.5625 + 1.5x + x2 = 15.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 15.5625 Calculate the square root of the right side: 3.94493346 Break this problem into two subproblems by setting (x + 0.75) equal to 3.94493346 and -3.94493346.Subproblem 1
x + 0.75 = 3.94493346 Simplifying x + 0.75 = 3.94493346 Reorder the terms: 0.75 + x = 3.94493346 Solving 0.75 + x = 3.94493346 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 3.94493346 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 3.94493346 + -0.75 x = 3.94493346 + -0.75 Combine like terms: 3.94493346 + -0.75 = 3.19493346 x = 3.19493346 Simplifying x = 3.19493346Subproblem 2
x + 0.75 = -3.94493346 Simplifying x + 0.75 = -3.94493346 Reorder the terms: 0.75 + x = -3.94493346 Solving 0.75 + x = -3.94493346 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -3.94493346 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -3.94493346 + -0.75 x = -3.94493346 + -0.75 Combine like terms: -3.94493346 + -0.75 = -4.69493346 x = -4.69493346 Simplifying x = -4.69493346Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.19493346, -4.69493346}
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