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Simplifying 6 = 16x2 + 24x + 1 Reorder the terms: 6 = 1 + 24x + 16x2 Solving 6 = 1 + 24x + 16x2 Solving for variable 'x'. Combine like terms: 6 + -1 = 5 5 + -24x + -16x2 = 1 + 24x + 16x2 + -1 + -24x + -16x2 Reorder the terms: 5 + -24x + -16x2 = 1 + -1 + 24x + -24x + 16x2 + -16x2 Combine like terms: 1 + -1 = 0 5 + -24x + -16x2 = 0 + 24x + -24x + 16x2 + -16x2 5 + -24x + -16x2 = 24x + -24x + 16x2 + -16x2 Combine like terms: 24x + -24x = 0 5 + -24x + -16x2 = 0 + 16x2 + -16x2 5 + -24x + -16x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 5 + -24x + -16x2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.3125 + 1.5x + x2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + 1.5x + 0.3125 + x2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + 1.5x + x2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + 1.5x + x2 = 0 + 0.3125 1.5x + x2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 1.5x + x2 = 0.3125 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 0.3125 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 0.3125 + 0.5625 Combine like terms: 0.3125 + 0.5625 = 0.875 0.5625 + 1.5x + x2 = 0.875 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 0.875 Calculate the square root of the right side: 0.935414347 Break this problem into two subproblems by setting (x + 0.75) equal to 0.935414347 and -0.935414347.Subproblem 1
x + 0.75 = 0.935414347 Simplifying x + 0.75 = 0.935414347 Reorder the terms: 0.75 + x = 0.935414347 Solving 0.75 + x = 0.935414347 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 0.935414347 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 0.935414347 + -0.75 x = 0.935414347 + -0.75 Combine like terms: 0.935414347 + -0.75 = 0.185414347 x = 0.185414347 Simplifying x = 0.185414347Subproblem 2
x + 0.75 = -0.935414347 Simplifying x + 0.75 = -0.935414347 Reorder the terms: 0.75 + x = -0.935414347 Solving 0.75 + x = -0.935414347 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -0.935414347 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -0.935414347 + -0.75 x = -0.935414347 + -0.75 Combine like terms: -0.935414347 + -0.75 = -1.685414347 x = -1.685414347 Simplifying x = -1.685414347Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.185414347, -1.685414347}
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