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Simplifying 23.76 = -0.5 * 32x2 + 12x + 22 Multiply -0.5 * 32 23.76 = -16x2 + 12x + 22 Reorder the terms: 23.76 = 22 + 12x + -16x2 Solving 23.76 = 22 + 12x + -16x2 Solving for variable 'x'. Combine like terms: 23.76 + -22 = 1.76 1.76 + -12x + 16x2 = 22 + 12x + -16x2 + -22 + -12x + 16x2 Reorder the terms: 1.76 + -12x + 16x2 = 22 + -22 + 12x + -12x + -16x2 + 16x2 Combine like terms: 22 + -22 = 0 1.76 + -12x + 16x2 = 0 + 12x + -12x + -16x2 + 16x2 1.76 + -12x + 16x2 = 12x + -12x + -16x2 + 16x2 Combine like terms: 12x + -12x = 0 1.76 + -12x + 16x2 = 0 + -16x2 + 16x2 1.76 + -12x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 1.76 + -12x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.11 + -0.75x + x2 = 0 Move the constant term to the right: Add '-0.11' to each side of the equation. 0.11 + -0.75x + -0.11 + x2 = 0 + -0.11 Reorder the terms: 0.11 + -0.11 + -0.75x + x2 = 0 + -0.11 Combine like terms: 0.11 + -0.11 = 0.00 0.00 + -0.75x + x2 = 0 + -0.11 -0.75x + x2 = 0 + -0.11 Combine like terms: 0 + -0.11 = -0.11 -0.75x + x2 = -0.11 The x term is -0.75x. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75x + 0.140625 + x2 = -0.11 + 0.140625 Reorder the terms: 0.140625 + -0.75x + x2 = -0.11 + 0.140625 Combine like terms: -0.11 + 0.140625 = 0.030625 0.140625 + -0.75x + x2 = 0.030625 Factor a perfect square on the left side: (x + -0.375)(x + -0.375) = 0.030625 Calculate the square root of the right side: 0.175 Break this problem into two subproblems by setting (x + -0.375) equal to 0.175 and -0.175.Subproblem 1
x + -0.375 = 0.175 Simplifying x + -0.375 = 0.175 Reorder the terms: -0.375 + x = 0.175 Solving -0.375 + x = 0.175 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 0.175 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 0.175 + 0.375 x = 0.175 + 0.375 Combine like terms: 0.175 + 0.375 = 0.55 x = 0.55 Simplifying x = 0.55Subproblem 2
x + -0.375 = -0.175 Simplifying x + -0.375 = -0.175 Reorder the terms: -0.375 + x = -0.175 Solving -0.375 + x = -0.175 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -0.175 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -0.175 + 0.375 x = -0.175 + 0.375 Combine like terms: -0.175 + 0.375 = 0.2 x = 0.2 Simplifying x = 0.2Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.55, 0.2}
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