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Simplifying (6x + 8)(9x + -25) = 180 Reorder the terms: (8 + 6x)(9x + -25) = 180 Reorder the terms: (8 + 6x)(-25 + 9x) = 180 Multiply (8 + 6x) * (-25 + 9x) (8(-25 + 9x) + 6x * (-25 + 9x)) = 180 ((-25 * 8 + 9x * 8) + 6x * (-25 + 9x)) = 180 ((-200 + 72x) + 6x * (-25 + 9x)) = 180 (-200 + 72x + (-25 * 6x + 9x * 6x)) = 180 (-200 + 72x + (-150x + 54x2)) = 180 Combine like terms: 72x + -150x = -78x (-200 + -78x + 54x2) = 180 Solving -200 + -78x + 54x2 = 180 Solving for variable 'x'. Reorder the terms: -200 + -180 + -78x + 54x2 = 180 + -180 Combine like terms: -200 + -180 = -380 -380 + -78x + 54x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -380 + -78x + 54x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-190 + -39x + 27x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-190 + -39x + 27x2)' equal to zero and attempt to solve: Simplifying -190 + -39x + 27x2 = 0 Solving -190 + -39x + 27x2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -7.037037037 + -1.444444444x + x2 = 0 Move the constant term to the right: Add '7.037037037' to each side of the equation. -7.037037037 + -1.444444444x + 7.037037037 + x2 = 0 + 7.037037037 Reorder the terms: -7.037037037 + 7.037037037 + -1.444444444x + x2 = 0 + 7.037037037 Combine like terms: -7.037037037 + 7.037037037 = 0.000000000 0.000000000 + -1.444444444x + x2 = 0 + 7.037037037 -1.444444444x + x2 = 0 + 7.037037037 Combine like terms: 0 + 7.037037037 = 7.037037037 -1.444444444x + x2 = 7.037037037 The x term is -1.444444444x. Take half its coefficient (-0.722222222). Square it (0.5216049380) and add it to both sides. Add '0.5216049380' to each side of the equation. -1.444444444x + 0.5216049380 + x2 = 7.037037037 + 0.5216049380 Reorder the terms: 0.5216049380 + -1.444444444x + x2 = 7.037037037 + 0.5216049380 Combine like terms: 7.037037037 + 0.5216049380 = 7.558641975 0.5216049380 + -1.444444444x + x2 = 7.558641975 Factor a perfect square on the left side: (x + -0.722222222)(x + -0.722222222) = 7.558641975 Calculate the square root of the right side: 2.749298451 Break this problem into two subproblems by setting (x + -0.722222222) equal to 2.749298451 and -2.749298451.Subproblem 1
x + -0.722222222 = 2.749298451 Simplifying x + -0.722222222 = 2.749298451 Reorder the terms: -0.722222222 + x = 2.749298451 Solving -0.722222222 + x = 2.749298451 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.722222222' to each side of the equation. -0.722222222 + 0.722222222 + x = 2.749298451 + 0.722222222 Combine like terms: -0.722222222 + 0.722222222 = 0.000000000 0.000000000 + x = 2.749298451 + 0.722222222 x = 2.749298451 + 0.722222222 Combine like terms: 2.749298451 + 0.722222222 = 3.471520673 x = 3.471520673 Simplifying x = 3.471520673Subproblem 2
x + -0.722222222 = -2.749298451 Simplifying x + -0.722222222 = -2.749298451 Reorder the terms: -0.722222222 + x = -2.749298451 Solving -0.722222222 + x = -2.749298451 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.722222222' to each side of the equation. -0.722222222 + 0.722222222 + x = -2.749298451 + 0.722222222 Combine like terms: -0.722222222 + 0.722222222 = 0.000000000 0.000000000 + x = -2.749298451 + 0.722222222 x = -2.749298451 + 0.722222222 Combine like terms: -2.749298451 + 0.722222222 = -2.027076229 x = -2.027076229 Simplifying x = -2.027076229Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.471520673, -2.027076229}Solution
x = {3.471520673, -2.027076229}
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