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Simplifying 15 = (3 + 2x)(3.25 + x) Multiply (3 + 2x) * (3.25 + x) 15 = (3(3.25 + x) + 2x * (3.25 + x)) 15 = ((3.25 * 3 + x * 3) + 2x * (3.25 + x)) 15 = ((9.75 + 3x) + 2x * (3.25 + x)) 15 = (9.75 + 3x + (3.25 * 2x + x * 2x)) 15 = (9.75 + 3x + (6.5x + 2x2)) Combine like terms: 3x + 6.5x = 9.5x 15 = (9.75 + 9.5x + 2x2) Solving 15 = 9.75 + 9.5x + 2x2 Solving for variable 'x'. Combine like terms: 15 + -9.75 = 5.25 5.25 + -9.5x + -2x2 = 9.75 + 9.5x + 2x2 + -9.75 + -9.5x + -2x2 Reorder the terms: 5.25 + -9.5x + -2x2 = 9.75 + -9.75 + 9.5x + -9.5x + 2x2 + -2x2 Combine like terms: 9.75 + -9.75 = 0.00 5.25 + -9.5x + -2x2 = 0.00 + 9.5x + -9.5x + 2x2 + -2x2 5.25 + -9.5x + -2x2 = 9.5x + -9.5x + 2x2 + -2x2 Combine like terms: 9.5x + -9.5x = 0.0 5.25 + -9.5x + -2x2 = 0.0 + 2x2 + -2x2 5.25 + -9.5x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 5.25 + -9.5x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.625 + 4.75x + x2 = 0 Move the constant term to the right: Add '2.625' to each side of the equation. -2.625 + 4.75x + 2.625 + x2 = 0 + 2.625 Reorder the terms: -2.625 + 2.625 + 4.75x + x2 = 0 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + 4.75x + x2 = 0 + 2.625 4.75x + x2 = 0 + 2.625 Combine like terms: 0 + 2.625 = 2.625 4.75x + x2 = 2.625 The x term is 4.75x. Take half its coefficient (2.375). Square it (5.640625) and add it to both sides. Add '5.640625' to each side of the equation. 4.75x + 5.640625 + x2 = 2.625 + 5.640625 Reorder the terms: 5.640625 + 4.75x + x2 = 2.625 + 5.640625 Combine like terms: 2.625 + 5.640625 = 8.265625 5.640625 + 4.75x + x2 = 8.265625 Factor a perfect square on the left side: (x + 2.375)(x + 2.375) = 8.265625 Calculate the square root of the right side: 2.875 Break this problem into two subproblems by setting (x + 2.375) equal to 2.875 and -2.875.Subproblem 1
x + 2.375 = 2.875 Simplifying x + 2.375 = 2.875 Reorder the terms: 2.375 + x = 2.875 Solving 2.375 + x = 2.875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + x = 2.875 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + x = 2.875 + -2.375 x = 2.875 + -2.375 Combine like terms: 2.875 + -2.375 = 0.5 x = 0.5 Simplifying x = 0.5Subproblem 2
x + 2.375 = -2.875 Simplifying x + 2.375 = -2.875 Reorder the terms: 2.375 + x = -2.875 Solving 2.375 + x = -2.875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + x = -2.875 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + x = -2.875 + -2.375 x = -2.875 + -2.375 Combine like terms: -2.875 + -2.375 = -5.25 x = -5.25 Simplifying x = -5.25Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.5, -5.25}
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