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Simplifying x(2x + 7) = 75 Reorder the terms: x(7 + 2x) = 75 (7 * x + 2x * x) = 75 (7x + 2x2) = 75 Solving 7x + 2x2 = 75 Solving for variable 'x'. Reorder the terms: -75 + 7x + 2x2 = 75 + -75 Combine like terms: 75 + -75 = 0 -75 + 7x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -37.5 + 3.5x + x2 = 0 Move the constant term to the right: Add '37.5' to each side of the equation. -37.5 + 3.5x + 37.5 + x2 = 0 + 37.5 Reorder the terms: -37.5 + 37.5 + 3.5x + x2 = 0 + 37.5 Combine like terms: -37.5 + 37.5 = 0.0 0.0 + 3.5x + x2 = 0 + 37.5 3.5x + x2 = 0 + 37.5 Combine like terms: 0 + 37.5 = 37.5 3.5x + x2 = 37.5 The x term is 3.5x. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5x + 3.0625 + x2 = 37.5 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = 37.5 + 3.0625 Combine like terms: 37.5 + 3.0625 = 40.5625 3.0625 + 3.5x + x2 = 40.5625 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 40.5625 Calculate the square root of the right side: 6.368869601 Break this problem into two subproblems by setting (x + 1.75) equal to 6.368869601 and -6.368869601.Subproblem 1
x + 1.75 = 6.368869601 Simplifying x + 1.75 = 6.368869601 Reorder the terms: 1.75 + x = 6.368869601 Solving 1.75 + x = 6.368869601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 6.368869601 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 6.368869601 + -1.75 x = 6.368869601 + -1.75 Combine like terms: 6.368869601 + -1.75 = 4.618869601 x = 4.618869601 Simplifying x = 4.618869601Subproblem 2
x + 1.75 = -6.368869601 Simplifying x + 1.75 = -6.368869601 Reorder the terms: 1.75 + x = -6.368869601 Solving 1.75 + x = -6.368869601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -6.368869601 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -6.368869601 + -1.75 x = -6.368869601 + -1.75 Combine like terms: -6.368869601 + -1.75 = -8.118869601 x = -8.118869601 Simplifying x = -8.118869601Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.618869601, -8.118869601}
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