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Simplifying 2x2 + 32x + 111 = 0 Reorder the terms: 111 + 32x + 2x2 = 0 Solving 111 + 32x + 2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 55.5 + 16x + x2 = 0 Move the constant term to the right: Add '-55.5' to each side of the equation. 55.5 + 16x + -55.5 + x2 = 0 + -55.5 Reorder the terms: 55.5 + -55.5 + 16x + x2 = 0 + -55.5 Combine like terms: 55.5 + -55.5 = 0.0 0.0 + 16x + x2 = 0 + -55.5 16x + x2 = 0 + -55.5 Combine like terms: 0 + -55.5 = -55.5 16x + x2 = -55.5 The x term is 16x. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16x + 64 + x2 = -55.5 + 64 Reorder the terms: 64 + 16x + x2 = -55.5 + 64 Combine like terms: -55.5 + 64 = 8.5 64 + 16x + x2 = 8.5 Factor a perfect square on the left side: (x + 8)(x + 8) = 8.5 Calculate the square root of the right side: 2.915475947 Break this problem into two subproblems by setting (x + 8) equal to 2.915475947 and -2.915475947.Subproblem 1
x + 8 = 2.915475947 Simplifying x + 8 = 2.915475947 Reorder the terms: 8 + x = 2.915475947 Solving 8 + x = 2.915475947 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + x = 2.915475947 + -8 Combine like terms: 8 + -8 = 0 0 + x = 2.915475947 + -8 x = 2.915475947 + -8 Combine like terms: 2.915475947 + -8 = -5.084524053 x = -5.084524053 Simplifying x = -5.084524053Subproblem 2
x + 8 = -2.915475947 Simplifying x + 8 = -2.915475947 Reorder the terms: 8 + x = -2.915475947 Solving 8 + x = -2.915475947 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + x = -2.915475947 + -8 Combine like terms: 8 + -8 = 0 0 + x = -2.915475947 + -8 x = -2.915475947 + -8 Combine like terms: -2.915475947 + -8 = -10.915475947 x = -10.915475947 Simplifying x = -10.915475947Solution
The solution to the problem is based on the solutions from the subproblems. x = {-5.084524053, -10.915475947}
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