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Simplifying (25 + 2x)(32 + 2x) = 310 Multiply (25 + 2x) * (32 + 2x) (25(32 + 2x) + 2x * (32 + 2x)) = 310 ((32 * 25 + 2x * 25) + 2x * (32 + 2x)) = 310 ((800 + 50x) + 2x * (32 + 2x)) = 310 (800 + 50x + (32 * 2x + 2x * 2x)) = 310 (800 + 50x + (64x + 4x2)) = 310 Combine like terms: 50x + 64x = 114x (800 + 114x + 4x2) = 310 Solving 800 + 114x + 4x2 = 310 Solving for variable 'x'. Reorder the terms: 800 + -310 + 114x + 4x2 = 310 + -310 Combine like terms: 800 + -310 = 490 490 + 114x + 4x2 = 310 + -310 Combine like terms: 310 + -310 = 0 490 + 114x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(245 + 57x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(245 + 57x + 2x2)' equal to zero and attempt to solve: Simplifying 245 + 57x + 2x2 = 0 Solving 245 + 57x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 122.5 + 28.5x + x2 = 0 Move the constant term to the right: Add '-122.5' to each side of the equation. 122.5 + 28.5x + -122.5 + x2 = 0 + -122.5 Reorder the terms: 122.5 + -122.5 + 28.5x + x2 = 0 + -122.5 Combine like terms: 122.5 + -122.5 = 0.0 0.0 + 28.5x + x2 = 0 + -122.5 28.5x + x2 = 0 + -122.5 Combine like terms: 0 + -122.5 = -122.5 28.5x + x2 = -122.5 The x term is 28.5x. Take half its coefficient (14.25). Square it (203.0625) and add it to both sides. Add '203.0625' to each side of the equation. 28.5x + 203.0625 + x2 = -122.5 + 203.0625 Reorder the terms: 203.0625 + 28.5x + x2 = -122.5 + 203.0625 Combine like terms: -122.5 + 203.0625 = 80.5625 203.0625 + 28.5x + x2 = 80.5625 Factor a perfect square on the left side: (x + 14.25)(x + 14.25) = 80.5625 Calculate the square root of the right side: 8.975661536 Break this problem into two subproblems by setting (x + 14.25) equal to 8.975661536 and -8.975661536.Subproblem 1
x + 14.25 = 8.975661536 Simplifying x + 14.25 = 8.975661536 Reorder the terms: 14.25 + x = 8.975661536 Solving 14.25 + x = 8.975661536 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-14.25' to each side of the equation. 14.25 + -14.25 + x = 8.975661536 + -14.25 Combine like terms: 14.25 + -14.25 = 0.00 0.00 + x = 8.975661536 + -14.25 x = 8.975661536 + -14.25 Combine like terms: 8.975661536 + -14.25 = -5.274338464 x = -5.274338464 Simplifying x = -5.274338464Subproblem 2
x + 14.25 = -8.975661536 Simplifying x + 14.25 = -8.975661536 Reorder the terms: 14.25 + x = -8.975661536 Solving 14.25 + x = -8.975661536 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-14.25' to each side of the equation. 14.25 + -14.25 + x = -8.975661536 + -14.25 Combine like terms: 14.25 + -14.25 = 0.00 0.00 + x = -8.975661536 + -14.25 x = -8.975661536 + -14.25 Combine like terms: -8.975661536 + -14.25 = -23.225661536 x = -23.225661536 Simplifying x = -23.225661536Solution
The solution to the problem is based on the solutions from the subproblems. x = {-5.274338464, -23.225661536}Solution
x = {-5.274338464, -23.225661536}
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