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Simplifying 15 = (2x + 3)(2x + 3.25) Reorder the terms: 15 = (3 + 2x)(2x + 3.25) Reorder the terms: 15 = (3 + 2x)(3.25 + 2x) Multiply (3 + 2x) * (3.25 + 2x) 15 = (3(3.25 + 2x) + 2x * (3.25 + 2x)) 15 = ((3.25 * 3 + 2x * 3) + 2x * (3.25 + 2x)) 15 = ((9.75 + 6x) + 2x * (3.25 + 2x)) 15 = (9.75 + 6x + (3.25 * 2x + 2x * 2x)) 15 = (9.75 + 6x + (6.5x + 4x2)) Combine like terms: 6x + 6.5x = 12.5x 15 = (9.75 + 12.5x + 4x2) Solving 15 = 9.75 + 12.5x + 4x2 Solving for variable 'x'. Combine like terms: 15 + -9.75 = 5.25 5.25 + -12.5x + -4x2 = 9.75 + 12.5x + 4x2 + -9.75 + -12.5x + -4x2 Reorder the terms: 5.25 + -12.5x + -4x2 = 9.75 + -9.75 + 12.5x + -12.5x + 4x2 + -4x2 Combine like terms: 9.75 + -9.75 = 0.00 5.25 + -12.5x + -4x2 = 0.00 + 12.5x + -12.5x + 4x2 + -4x2 5.25 + -12.5x + -4x2 = 12.5x + -12.5x + 4x2 + -4x2 Combine like terms: 12.5x + -12.5x = 0.0 5.25 + -12.5x + -4x2 = 0.0 + 4x2 + -4x2 5.25 + -12.5x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 5.25 + -12.5x + -4x2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.3125 + 3.125x + x2 = 0 Move the constant term to the right: Add '1.3125' to each side of the equation. -1.3125 + 3.125x + 1.3125 + x2 = 0 + 1.3125 Reorder the terms: -1.3125 + 1.3125 + 3.125x + x2 = 0 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + 3.125x + x2 = 0 + 1.3125 3.125x + x2 = 0 + 1.3125 Combine like terms: 0 + 1.3125 = 1.3125 3.125x + x2 = 1.3125 The x term is 3.125x. Take half its coefficient (1.5625). Square it (2.44140625) and add it to both sides. Add '2.44140625' to each side of the equation. 3.125x + 2.44140625 + x2 = 1.3125 + 2.44140625 Reorder the terms: 2.44140625 + 3.125x + x2 = 1.3125 + 2.44140625 Combine like terms: 1.3125 + 2.44140625 = 3.75390625 2.44140625 + 3.125x + x2 = 3.75390625 Factor a perfect square on the left side: (x + 1.5625)(x + 1.5625) = 3.75390625 Calculate the square root of the right side: 1.9375 Break this problem into two subproblems by setting (x + 1.5625) equal to 1.9375 and -1.9375.Subproblem 1
x + 1.5625 = 1.9375 Simplifying x + 1.5625 = 1.9375 Reorder the terms: 1.5625 + x = 1.9375 Solving 1.5625 + x = 1.9375 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5625' to each side of the equation. 1.5625 + -1.5625 + x = 1.9375 + -1.5625 Combine like terms: 1.5625 + -1.5625 = 0.0000 0.0000 + x = 1.9375 + -1.5625 x = 1.9375 + -1.5625 Combine like terms: 1.9375 + -1.5625 = 0.375 x = 0.375 Simplifying x = 0.375Subproblem 2
x + 1.5625 = -1.9375 Simplifying x + 1.5625 = -1.9375 Reorder the terms: 1.5625 + x = -1.9375 Solving 1.5625 + x = -1.9375 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5625' to each side of the equation. 1.5625 + -1.5625 + x = -1.9375 + -1.5625 Combine like terms: 1.5625 + -1.5625 = 0.0000 0.0000 + x = -1.9375 + -1.5625 x = -1.9375 + -1.5625 Combine like terms: -1.9375 + -1.5625 = -3.5 x = -3.5 Simplifying x = -3.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.375, -3.5}
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