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Simplifying 0 = -16x2 + 120x + 3 Reorder the terms: 0 = 3 + 120x + -16x2 Solving 0 = 3 + 120x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -3 = -3 -3 + -120x + 16x2 = 3 + 120x + -16x2 + -3 + -120x + 16x2 Reorder the terms: -3 + -120x + 16x2 = 3 + -3 + 120x + -120x + -16x2 + 16x2 Combine like terms: 3 + -3 = 0 -3 + -120x + 16x2 = 0 + 120x + -120x + -16x2 + 16x2 -3 + -120x + 16x2 = 120x + -120x + -16x2 + 16x2 Combine like terms: 120x + -120x = 0 -3 + -120x + 16x2 = 0 + -16x2 + 16x2 -3 + -120x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -3 + -120x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.1875 + -7.5x + x2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -7.5x + 0.1875 + x2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -7.5x + x2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -7.5x + x2 = 0 + 0.1875 -7.5x + x2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -7.5x + x2 = 0.1875 The x term is -7.5x. Take half its coefficient (-3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. -7.5x + 14.0625 + x2 = 0.1875 + 14.0625 Reorder the terms: 14.0625 + -7.5x + x2 = 0.1875 + 14.0625 Combine like terms: 0.1875 + 14.0625 = 14.25 14.0625 + -7.5x + x2 = 14.25 Factor a perfect square on the left side: (x + -3.75)(x + -3.75) = 14.25 Calculate the square root of the right side: 3.774917218 Break this problem into two subproblems by setting (x + -3.75) equal to 3.774917218 and -3.774917218.Subproblem 1
x + -3.75 = 3.774917218 Simplifying x + -3.75 = 3.774917218 Reorder the terms: -3.75 + x = 3.774917218 Solving -3.75 + x = 3.774917218 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + x = 3.774917218 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + x = 3.774917218 + 3.75 x = 3.774917218 + 3.75 Combine like terms: 3.774917218 + 3.75 = 7.524917218 x = 7.524917218 Simplifying x = 7.524917218Subproblem 2
x + -3.75 = -3.774917218 Simplifying x + -3.75 = -3.774917218 Reorder the terms: -3.75 + x = -3.774917218 Solving -3.75 + x = -3.774917218 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + x = -3.774917218 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + x = -3.774917218 + 3.75 x = -3.774917218 + 3.75 Combine like terms: -3.774917218 + 3.75 = -0.024917218 x = -0.024917218 Simplifying x = -0.024917218Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.524917218, -0.024917218}
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