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Simplifying (x2 + -5x + 6)(x + -1) = (x2 + 8x + 16)(x + -4) + 7 Reorder the terms: (6 + -5x + x2)(x + -1) = (x2 + 8x + 16)(x + -4) + 7 Reorder the terms: (6 + -5x + x2)(-1 + x) = (x2 + 8x + 16)(x + -4) + 7 Multiply (6 + -5x + x2) * (-1 + x) (6(-1 + x) + -5x * (-1 + x) + x2(-1 + x)) = (x2 + 8x + 16)(x + -4) + 7 ((-1 * 6 + x * 6) + -5x * (-1 + x) + x2(-1 + x)) = (x2 + 8x + 16)(x + -4) + 7 ((-6 + 6x) + -5x * (-1 + x) + x2(-1 + x)) = (x2 + 8x + 16)(x + -4) + 7 (-6 + 6x + (-1 * -5x + x * -5x) + x2(-1 + x)) = (x2 + 8x + 16)(x + -4) + 7 (-6 + 6x + (5x + -5x2) + x2(-1 + x)) = (x2 + 8x + 16)(x + -4) + 7 (-6 + 6x + 5x + -5x2 + (-1 * x2 + x * x2)) = (x2 + 8x + 16)(x + -4) + 7 (-6 + 6x + 5x + -5x2 + (-1x2 + x3)) = (x2 + 8x + 16)(x + -4) + 7 Combine like terms: 6x + 5x = 11x (-6 + 11x + -5x2 + -1x2 + x3) = (x2 + 8x + 16)(x + -4) + 7 Combine like terms: -5x2 + -1x2 = -6x2 (-6 + 11x + -6x2 + x3) = (x2 + 8x + 16)(x + -4) + 7 Reorder the terms: -6 + 11x + -6x2 + x3 = (16 + 8x + x2)(x + -4) + 7 Reorder the terms: -6 + 11x + -6x2 + x3 = (16 + 8x + x2)(-4 + x) + 7 Multiply (16 + 8x + x2) * (-4 + x) -6 + 11x + -6x2 + x3 = (16(-4 + x) + 8x * (-4 + x) + x2(-4 + x)) + 7 -6 + 11x + -6x2 + x3 = ((-4 * 16 + x * 16) + 8x * (-4 + x) + x2(-4 + x)) + 7 -6 + 11x + -6x2 + x3 = ((-64 + 16x) + 8x * (-4 + x) + x2(-4 + x)) + 7 -6 + 11x + -6x2 + x3 = (-64 + 16x + (-4 * 8x + x * 8x) + x2(-4 + x)) + 7 -6 + 11x + -6x2 + x3 = (-64 + 16x + (-32x + 8x2) + x2(-4 + x)) + 7 -6 + 11x + -6x2 + x3 = (-64 + 16x + -32x + 8x2 + (-4 * x2 + x * x2)) + 7 -6 + 11x + -6x2 + x3 = (-64 + 16x + -32x + 8x2 + (-4x2 + x3)) + 7 Combine like terms: 16x + -32x = -16x -6 + 11x + -6x2 + x3 = (-64 + -16x + 8x2 + -4x2 + x3) + 7 Combine like terms: 8x2 + -4x2 = 4x2 -6 + 11x + -6x2 + x3 = (-64 + -16x + 4x2 + x3) + 7 Reorder the terms: -6 + 11x + -6x2 + x3 = -64 + 7 + -16x + 4x2 + x3 Combine like terms: -64 + 7 = -57 -6 + 11x + -6x2 + x3 = -57 + -16x + 4x2 + x3 Add '-1x3' to each side of the equation. -6 + 11x + -6x2 + x3 + -1x3 = -57 + -16x + 4x2 + x3 + -1x3 Combine like terms: x3 + -1x3 = 0 -6 + 11x + -6x2 + 0 = -57 + -16x + 4x2 + x3 + -1x3 -6 + 11x + -6x2 = -57 + -16x + 4x2 + x3 + -1x3 Combine like terms: x3 + -1x3 = 0 -6 + 11x + -6x2 = -57 + -16x + 4x2 + 0 -6 + 11x + -6x2 = -57 + -16x + 4x2 Solving -6 + 11x + -6x2 = -57 + -16x + 4x2 Solving for variable 'x'. Reorder the terms: -6 + 57 + 11x + 16x + -6x2 + -4x2 = -57 + -16x + 4x2 + 57 + 16x + -4x2 Combine like terms: -6 + 57 = 51 51 + 11x + 16x + -6x2 + -4x2 = -57 + -16x + 4x2 + 57 + 16x + -4x2 Combine like terms: 11x + 16x = 27x 51 + 27x + -6x2 + -4x2 = -57 + -16x + 4x2 + 57 + 16x + -4x2 Combine like terms: -6x2 + -4x2 = -10x2 51 + 27x + -10x2 = -57 + -16x + 4x2 + 57 + 16x + -4x2 Reorder the terms: 51 + 27x + -10x2 = -57 + 57 + -16x + 16x + 4x2 + -4x2 Combine like terms: -57 + 57 = 0 51 + 27x + -10x2 = 0 + -16x + 16x + 4x2 + -4x2 51 + 27x + -10x2 = -16x + 16x + 4x2 + -4x2 Combine like terms: -16x + 16x = 0 51 + 27x + -10x2 = 0 + 4x2 + -4x2 51 + 27x + -10x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 51 + 27x + -10x2 = 0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -5.1 + -2.7x + x2 = 0.0 Move the constant term to the right: Add '5.1' to each side of the equation. -5.1 + -2.7x + 5.1 + x2 = 0.0 + 5.1 Reorder the terms: -5.1 + 5.1 + -2.7x + x2 = 0.0 + 5.1 Combine like terms: -5.1 + 5.1 = 0.0 0.0 + -2.7x + x2 = 0.0 + 5.1 -2.7x + x2 = 0.0 + 5.1 Combine like terms: 0.0 + 5.1 = 5.1 -2.7x + x2 = 5.1 The x term is -2.7x. Take half its coefficient (-1.35). Square it (1.8225) and add it to both sides. Add '1.8225' to each side of the equation. -2.7x + 1.8225 + x2 = 5.1 + 1.8225 Reorder the terms: 1.8225 + -2.7x + x2 = 5.1 + 1.8225 Combine like terms: 5.1 + 1.8225 = 6.9225 1.8225 + -2.7x + x2 = 6.9225 Factor a perfect square on the left side: (x + -1.35)(x + -1.35) = 6.9225 Calculate the square root of the right side: 2.631064423 Break this problem into two subproblems by setting (x + -1.35) equal to 2.631064423 and -2.631064423.Subproblem 1
x + -1.35 = 2.631064423 Simplifying x + -1.35 = 2.631064423 Reorder the terms: -1.35 + x = 2.631064423 Solving -1.35 + x = 2.631064423 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.35' to each side of the equation. -1.35 + 1.35 + x = 2.631064423 + 1.35 Combine like terms: -1.35 + 1.35 = 0.00 0.00 + x = 2.631064423 + 1.35 x = 2.631064423 + 1.35 Combine like terms: 2.631064423 + 1.35 = 3.981064423 x = 3.981064423 Simplifying x = 3.981064423Subproblem 2
x + -1.35 = -2.631064423 Simplifying x + -1.35 = -2.631064423 Reorder the terms: -1.35 + x = -2.631064423 Solving -1.35 + x = -2.631064423 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.35' to each side of the equation. -1.35 + 1.35 + x = -2.631064423 + 1.35 Combine like terms: -1.35 + 1.35 = 0.00 0.00 + x = -2.631064423 + 1.35 x = -2.631064423 + 1.35 Combine like terms: -2.631064423 + 1.35 = -1.281064423 x = -1.281064423 Simplifying x = -1.281064423Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.981064423, -1.281064423}
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