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Simplifying (3x + -12)(2x + 5) = 196 Reorder the terms: (-12 + 3x)(2x + 5) = 196 Reorder the terms: (-12 + 3x)(5 + 2x) = 196 Multiply (-12 + 3x) * (5 + 2x) (-12(5 + 2x) + 3x * (5 + 2x)) = 196 ((5 * -12 + 2x * -12) + 3x * (5 + 2x)) = 196 ((-60 + -24x) + 3x * (5 + 2x)) = 196 (-60 + -24x + (5 * 3x + 2x * 3x)) = 196 (-60 + -24x + (15x + 6x2)) = 196 Combine like terms: -24x + 15x = -9x (-60 + -9x + 6x2) = 196 Solving -60 + -9x + 6x2 = 196 Solving for variable 'x'. Reorder the terms: -60 + -196 + -9x + 6x2 = 196 + -196 Combine like terms: -60 + -196 = -256 -256 + -9x + 6x2 = 196 + -196 Combine like terms: 196 + -196 = 0 -256 + -9x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -42.66666667 + -1.5x + x2 = 0 Move the constant term to the right: Add '42.66666667' to each side of the equation. -42.66666667 + -1.5x + 42.66666667 + x2 = 0 + 42.66666667 Reorder the terms: -42.66666667 + 42.66666667 + -1.5x + x2 = 0 + 42.66666667 Combine like terms: -42.66666667 + 42.66666667 = 0.00000000 0.00000000 + -1.5x + x2 = 0 + 42.66666667 -1.5x + x2 = 0 + 42.66666667 Combine like terms: 0 + 42.66666667 = 42.66666667 -1.5x + x2 = 42.66666667 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = 42.66666667 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = 42.66666667 + 0.5625 Combine like terms: 42.66666667 + 0.5625 = 43.22916667 0.5625 + -1.5x + x2 = 43.22916667 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 43.22916667 Calculate the square root of the right side: 6.574889099 Break this problem into two subproblems by setting (x + -0.75) equal to 6.574889099 and -6.574889099.Subproblem 1
x + -0.75 = 6.574889099 Simplifying x + -0.75 = 6.574889099 Reorder the terms: -0.75 + x = 6.574889099 Solving -0.75 + x = 6.574889099 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 6.574889099 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 6.574889099 + 0.75 x = 6.574889099 + 0.75 Combine like terms: 6.574889099 + 0.75 = 7.324889099 x = 7.324889099 Simplifying x = 7.324889099Subproblem 2
x + -0.75 = -6.574889099 Simplifying x + -0.75 = -6.574889099 Reorder the terms: -0.75 + x = -6.574889099 Solving -0.75 + x = -6.574889099 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -6.574889099 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -6.574889099 + 0.75 x = -6.574889099 + 0.75 Combine like terms: -6.574889099 + 0.75 = -5.824889099 x = -5.824889099 Simplifying x = -5.824889099Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.324889099, -5.824889099}
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