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Simplifying 0.5(x + 8) * 1.5x = 10 Reorder the terms: 0.5(8 + x) * 1.5x = 10 Reorder the terms for easier multiplication: 0.5 * 1.5x(8 + x) = 10 Multiply 0.5 * 1.5 0.75x(8 + x) = 10 (8 * 0.75x + x * 0.75x) = 10 (6x + 0.75x2) = 10 Solving 6x + 0.75x2 = 10 Solving for variable 'x'. Reorder the terms: -10 + 6x + 0.75x2 = 10 + -10 Combine like terms: 10 + -10 = 0 -10 + 6x + 0.75x2 = 0 Begin completing the square. Divide all terms by 0.75 the coefficient of the squared term: Divide each side by '0.75'. -13.33333333 + 8x + x2 = 0 Move the constant term to the right: Add '13.33333333' to each side of the equation. -13.33333333 + 8x + 13.33333333 + x2 = 0 + 13.33333333 Reorder the terms: -13.33333333 + 13.33333333 + 8x + x2 = 0 + 13.33333333 Combine like terms: -13.33333333 + 13.33333333 = 0.00000000 0.00000000 + 8x + x2 = 0 + 13.33333333 8x + x2 = 0 + 13.33333333 Combine like terms: 0 + 13.33333333 = 13.33333333 8x + x2 = 13.33333333 The x term is 8x. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8x + 16 + x2 = 13.33333333 + 16 Reorder the terms: 16 + 8x + x2 = 13.33333333 + 16 Combine like terms: 13.33333333 + 16 = 29.33333333 16 + 8x + x2 = 29.33333333 Factor a perfect square on the left side: (x + 4)(x + 4) = 29.33333333 Calculate the square root of the right side: 5.416025603 Break this problem into two subproblems by setting (x + 4) equal to 5.416025603 and -5.416025603.Subproblem 1
x + 4 = 5.416025603 Simplifying x + 4 = 5.416025603 Reorder the terms: 4 + x = 5.416025603 Solving 4 + x = 5.416025603 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = 5.416025603 + -4 Combine like terms: 4 + -4 = 0 0 + x = 5.416025603 + -4 x = 5.416025603 + -4 Combine like terms: 5.416025603 + -4 = 1.416025603 x = 1.416025603 Simplifying x = 1.416025603Subproblem 2
x + 4 = -5.416025603 Simplifying x + 4 = -5.416025603 Reorder the terms: 4 + x = -5.416025603 Solving 4 + x = -5.416025603 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = -5.416025603 + -4 Combine like terms: 4 + -4 = 0 0 + x = -5.416025603 + -4 x = -5.416025603 + -4 Combine like terms: -5.416025603 + -4 = -9.416025603 x = -9.416025603 Simplifying x = -9.416025603Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.416025603, -9.416025603}
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