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Simplifying 8(X + 1)(X + -1) = 4(X + 1) + 27 Reorder the terms: 8(1 + X)(X + -1) = 4(X + 1) + 27 Reorder the terms: 8(1 + X)(-1 + X) = 4(X + 1) + 27 Multiply (1 + X) * (-1 + X) 8(1(-1 + X) + X(-1 + X)) = 4(X + 1) + 27 8((-1 * 1 + X * 1) + X(-1 + X)) = 4(X + 1) + 27 8((-1 + 1X) + X(-1 + X)) = 4(X + 1) + 27 8(-1 + 1X + (-1 * X + X * X)) = 4(X + 1) + 27 8(-1 + 1X + (-1X + X2)) = 4(X + 1) + 27 Combine like terms: 1X + -1X = 0 8(-1 + 0 + X2) = 4(X + 1) + 27 8(-1 + X2) = 4(X + 1) + 27 (-1 * 8 + X2 * 8) = 4(X + 1) + 27 (-8 + 8X2) = 4(X + 1) + 27 Reorder the terms: -8 + 8X2 = 4(1 + X) + 27 -8 + 8X2 = (1 * 4 + X * 4) + 27 -8 + 8X2 = (4 + 4X) + 27 Reorder the terms: -8 + 8X2 = 4 + 27 + 4X Combine like terms: 4 + 27 = 31 -8 + 8X2 = 31 + 4X Solving -8 + 8X2 = 31 + 4X Solving for variable 'X'. Reorder the terms: -8 + -31 + -4X + 8X2 = 31 + 4X + -31 + -4X Combine like terms: -8 + -31 = -39 -39 + -4X + 8X2 = 31 + 4X + -31 + -4X Reorder the terms: -39 + -4X + 8X2 = 31 + -31 + 4X + -4X Combine like terms: 31 + -31 = 0 -39 + -4X + 8X2 = 0 + 4X + -4X -39 + -4X + 8X2 = 4X + -4X Combine like terms: 4X + -4X = 0 -39 + -4X + 8X2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -4.875 + -0.5X + X2 = 0 Move the constant term to the right: Add '4.875' to each side of the equation. -4.875 + -0.5X + 4.875 + X2 = 0 + 4.875 Reorder the terms: -4.875 + 4.875 + -0.5X + X2 = 0 + 4.875 Combine like terms: -4.875 + 4.875 = 0.000 0.000 + -0.5X + X2 = 0 + 4.875 -0.5X + X2 = 0 + 4.875 Combine like terms: 0 + 4.875 = 4.875 -0.5X + X2 = 4.875 The X term is -0.5X. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5X + 0.0625 + X2 = 4.875 + 0.0625 Reorder the terms: 0.0625 + -0.5X + X2 = 4.875 + 0.0625 Combine like terms: 4.875 + 0.0625 = 4.9375 0.0625 + -0.5X + X2 = 4.9375 Factor a perfect square on the left side: (X + -0.25)(X + -0.25) = 4.9375 Calculate the square root of the right side: 2.222048604 Break this problem into two subproblems by setting (X + -0.25) equal to 2.222048604 and -2.222048604.Subproblem 1
X + -0.25 = 2.222048604 Simplifying X + -0.25 = 2.222048604 Reorder the terms: -0.25 + X = 2.222048604 Solving -0.25 + X = 2.222048604 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + X = 2.222048604 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + X = 2.222048604 + 0.25 X = 2.222048604 + 0.25 Combine like terms: 2.222048604 + 0.25 = 2.472048604 X = 2.472048604 Simplifying X = 2.472048604Subproblem 2
X + -0.25 = -2.222048604 Simplifying X + -0.25 = -2.222048604 Reorder the terms: -0.25 + X = -2.222048604 Solving -0.25 + X = -2.222048604 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + X = -2.222048604 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + X = -2.222048604 + 0.25 X = -2.222048604 + 0.25 Combine like terms: -2.222048604 + 0.25 = -1.972048604 X = -1.972048604 Simplifying X = -1.972048604Solution
The solution to the problem is based on the solutions from the subproblems. X = {2.472048604, -1.972048604}
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