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Simplifying 0.5(x + 3)(x) = 35 Reorder the terms: 0.5(3 + x)(x) = 35 Reorder the terms for easier multiplication: 0.5x(3 + x) = 35 (3 * 0.5x + x * 0.5x) = 35 (1.5x + 0.5x2) = 35 Solving 1.5x + 0.5x2 = 35 Solving for variable 'x'. Reorder the terms: -35 + 1.5x + 0.5x2 = 35 + -35 Combine like terms: 35 + -35 = 0 -35 + 1.5x + 0.5x2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -70 + 3x + x2 = 0 Move the constant term to the right: Add '70' to each side of the equation. -70 + 3x + 70 + x2 = 0 + 70 Reorder the terms: -70 + 70 + 3x + x2 = 0 + 70 Combine like terms: -70 + 70 = 0 0 + 3x + x2 = 0 + 70 3x + x2 = 0 + 70 Combine like terms: 0 + 70 = 70 3x + x2 = 70 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 = 70 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 70 + 2.25 Combine like terms: 70 + 2.25 = 72.25 2.25 + 3x + x2 = 72.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 72.25 Calculate the square root of the right side: 8.5 Break this problem into two subproblems by setting (x + 1.5) equal to 8.5 and -8.5.Subproblem 1
x + 1.5 = 8.5 Simplifying x + 1.5 = 8.5 Reorder the terms: 1.5 + x = 8.5 Solving 1.5 + x = 8.5 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 = 8.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 8.5 + -1.5 x = 8.5 + -1.5 Combine like terms: 8.5 + -1.5 = 7 x = 7 Simplifying x = 7Subproblem 2
x + 1.5 = -8.5 Simplifying x + 1.5 = -8.5 Reorder the terms: 1.5 + x = -8.5 Solving 1.5 + x = -8.5 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 = -8.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -8.5 + -1.5 x = -8.5 + -1.5 Combine like terms: -8.5 + -1.5 = -10 x = -10 Simplifying x = -10Solution
The solution to the problem is based on the solutions from the subproblems. x = {7, -10}
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