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Simplifying x * 1.5x + 3x = 120 Reorder the terms for easier multiplication: 1.5x * x + 3x = 120 Multiply x * x 1.5x2 + 3x = 120 Reorder the terms: 3x + 1.5x2 = 120 Solving 3x + 1.5x2 = 120 Solving for variable 'x'. Reorder the terms: -120 + 3x + 1.5x2 = 120 + -120 Combine like terms: 120 + -120 = 0 -120 + 3x + 1.5x2 = 0 Begin completing the square. Divide all terms by 1.5 the coefficient of the squared term: Divide each side by '1.5'. -80 + 2x + x2 = 0 Move the constant term to the right: Add '80' to each side of the equation. -80 + 2x + 80 + x2 = 0 + 80 Reorder the terms: -80 + 80 + 2x + x2 = 0 + 80 Combine like terms: -80 + 80 = 0 0 + 2x + x2 = 0 + 80 2x + x2 = 0 + 80 Combine like terms: 0 + 80 = 80 2x + x2 = 80 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 80 + 1 Reorder the terms: 1 + 2x + x2 = 80 + 1 Combine like terms: 80 + 1 = 81 1 + 2x + x2 = 81 Factor a perfect square on the left side: (x + 1)(x + 1) = 81 Calculate the square root of the right side: 9 Break this problem into two subproblems by setting (x + 1) equal to 9 and -9.Subproblem 1
x + 1 = 9 Simplifying x + 1 = 9 Reorder the terms: 1 + x = 9 Solving 1 + x = 9 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 9 + -1 Combine like terms: 1 + -1 = 0 0 + x = 9 + -1 x = 9 + -1 Combine like terms: 9 + -1 = 8 x = 8 Simplifying x = 8Subproblem 2
x + 1 = -9 Simplifying x + 1 = -9 Reorder the terms: 1 + x = -9 Solving 1 + x = -9 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -9 + -1 Combine like terms: 1 + -1 = 0 0 + x = -9 + -1 x = -9 + -1 Combine like terms: -9 + -1 = -10 x = -10 Simplifying x = -10Solution
The solution to the problem is based on the solutions from the subproblems. x = {8, -10}
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