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Simplifying 0 = 27x2 + 32x + -32 Reorder the terms: 0 = -32 + 32x + 27x2 Solving 0 = -32 + 32x + 27x2 Solving for variable 'x'. Combine like terms: 0 + 32 = 32 32 + -32x + -27x2 = -32 + 32x + 27x2 + 32 + -32x + -27x2 Reorder the terms: 32 + -32x + -27x2 = -32 + 32 + 32x + -32x + 27x2 + -27x2 Combine like terms: -32 + 32 = 0 32 + -32x + -27x2 = 0 + 32x + -32x + 27x2 + -27x2 32 + -32x + -27x2 = 32x + -32x + 27x2 + -27x2 Combine like terms: 32x + -32x = 0 32 + -32x + -27x2 = 0 + 27x2 + -27x2 32 + -32x + -27x2 = 27x2 + -27x2 Combine like terms: 27x2 + -27x2 = 0 32 + -32x + -27x2 = 0 Begin completing the square. Divide all terms by -27 the coefficient of the squared term: Divide each side by '-27'. -1.185185185 + 1.185185185x + x2 = 0 Move the constant term to the right: Add '1.185185185' to each side of the equation. -1.185185185 + 1.185185185x + 1.185185185 + x2 = 0 + 1.185185185 Reorder the terms: -1.185185185 + 1.185185185 + 1.185185185x + x2 = 0 + 1.185185185 Combine like terms: -1.185185185 + 1.185185185 = 0.000000000 0.000000000 + 1.185185185x + x2 = 0 + 1.185185185 1.185185185x + x2 = 0 + 1.185185185 Combine like terms: 0 + 1.185185185 = 1.185185185 1.185185185x + x2 = 1.185185185 The x term is 1.185185185x. Take half its coefficient (0.5925925925). Square it (0.3511659807) and add it to both sides. Add '0.3511659807' to each side of the equation. 1.185185185x + 0.3511659807 + x2 = 1.185185185 + 0.3511659807 Reorder the terms: 0.3511659807 + 1.185185185x + x2 = 1.185185185 + 0.3511659807 Combine like terms: 1.185185185 + 0.3511659807 = 1.5363511657 0.3511659807 + 1.185185185x + x2 = 1.5363511657 Factor a perfect square on the left side: (x + 0.5925925925)(x + 0.5925925925) = 1.5363511657 Calculate the square root of the right side: 1.239496335 Break this problem into two subproblems by setting (x + 0.5925925925) equal to 1.239496335 and -1.239496335.Subproblem 1
x + 0.5925925925 = 1.239496335 Simplifying x + 0.5925925925 = 1.239496335 Reorder the terms: 0.5925925925 + x = 1.239496335 Solving 0.5925925925 + x = 1.239496335 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5925925925' to each side of the equation. 0.5925925925 + -0.5925925925 + x = 1.239496335 + -0.5925925925 Combine like terms: 0.5925925925 + -0.5925925925 = 0.0000000000 0.0000000000 + x = 1.239496335 + -0.5925925925 x = 1.239496335 + -0.5925925925 Combine like terms: 1.239496335 + -0.5925925925 = 0.6469037425 x = 0.6469037425 Simplifying x = 0.6469037425Subproblem 2
x + 0.5925925925 = -1.239496335 Simplifying x + 0.5925925925 = -1.239496335 Reorder the terms: 0.5925925925 + x = -1.239496335 Solving 0.5925925925 + x = -1.239496335 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5925925925' to each side of the equation. 0.5925925925 + -0.5925925925 + x = -1.239496335 + -0.5925925925 Combine like terms: 0.5925925925 + -0.5925925925 = 0.0000000000 0.0000000000 + x = -1.239496335 + -0.5925925925 x = -1.239496335 + -0.5925925925 Combine like terms: -1.239496335 + -0.5925925925 = -1.8320889275 x = -1.8320889275 Simplifying x = -1.8320889275Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.6469037425, -1.8320889275}
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