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Simplifying 3(4x + -2) * 6(4x + -7) = 452 Reorder the terms: 3(-2 + 4x) * 6(4x + -7) = 452 Reorder the terms: 3(-2 + 4x) * 6(-7 + 4x) = 452 Reorder the terms for easier multiplication: 3 * 6(-2 + 4x)(-7 + 4x) = 452 Multiply 3 * 6 18(-2 + 4x)(-7 + 4x) = 452 Multiply (-2 + 4x) * (-7 + 4x) 18(-2(-7 + 4x) + 4x * (-7 + 4x)) = 452 18((-7 * -2 + 4x * -2) + 4x * (-7 + 4x)) = 452 18((14 + -8x) + 4x * (-7 + 4x)) = 452 18(14 + -8x + (-7 * 4x + 4x * 4x)) = 452 18(14 + -8x + (-28x + 16x2)) = 452 Combine like terms: -8x + -28x = -36x 18(14 + -36x + 16x2) = 452 (14 * 18 + -36x * 18 + 16x2 * 18) = 452 (252 + -648x + 288x2) = 452 Solving 252 + -648x + 288x2 = 452 Solving for variable 'x'. Reorder the terms: 252 + -452 + -648x + 288x2 = 452 + -452 Combine like terms: 252 + -452 = -200 -200 + -648x + 288x2 = 452 + -452 Combine like terms: 452 + -452 = 0 -200 + -648x + 288x2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-25 + -81x + 36x2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-25 + -81x + 36x2)' equal to zero and attempt to solve: Simplifying -25 + -81x + 36x2 = 0 Solving -25 + -81x + 36x2 = 0 Begin completing the square. Divide all terms by 36 the coefficient of the squared term: Divide each side by '36'. -0.6944444444 + -2.25x + x2 = 0 Move the constant term to the right: Add '0.6944444444' to each side of the equation. -0.6944444444 + -2.25x + 0.6944444444 + x2 = 0 + 0.6944444444 Reorder the terms: -0.6944444444 + 0.6944444444 + -2.25x + x2 = 0 + 0.6944444444 Combine like terms: -0.6944444444 + 0.6944444444 = 0.0000000000 0.0000000000 + -2.25x + x2 = 0 + 0.6944444444 -2.25x + x2 = 0 + 0.6944444444 Combine like terms: 0 + 0.6944444444 = 0.6944444444 -2.25x + x2 = 0.6944444444 The x term is -2.25x. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25x + 1.265625 + x2 = 0.6944444444 + 1.265625 Reorder the terms: 1.265625 + -2.25x + x2 = 0.6944444444 + 1.265625 Combine like terms: 0.6944444444 + 1.265625 = 1.9600694444 1.265625 + -2.25x + x2 = 1.9600694444 Factor a perfect square on the left side: (x + -1.125)(x + -1.125) = 1.9600694444 Calculate the square root of the right side: 1.400024801 Break this problem into two subproblems by setting (x + -1.125) equal to 1.400024801 and -1.400024801.Subproblem 1
x + -1.125 = 1.400024801 Simplifying x + -1.125 = 1.400024801 Reorder the terms: -1.125 + x = 1.400024801 Solving -1.125 + x = 1.400024801 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + x = 1.400024801 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + x = 1.400024801 + 1.125 x = 1.400024801 + 1.125 Combine like terms: 1.400024801 + 1.125 = 2.525024801 x = 2.525024801 Simplifying x = 2.525024801Subproblem 2
x + -1.125 = -1.400024801 Simplifying x + -1.125 = -1.400024801 Reorder the terms: -1.125 + x = -1.400024801 Solving -1.125 + x = -1.400024801 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + x = -1.400024801 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + x = -1.400024801 + 1.125 x = -1.400024801 + 1.125 Combine like terms: -1.400024801 + 1.125 = -0.275024801 x = -0.275024801 Simplifying x = -0.275024801Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.525024801, -0.275024801}Solution
x = {2.525024801, -0.275024801}
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