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Simplifying 16x2 + 40x = -11 Reorder the terms: 40x + 16x2 = -11 Solving 40x + 16x2 = -11 Solving for variable 'x'. Reorder the terms: 11 + 40x + 16x2 = -11 + 11 Combine like terms: -11 + 11 = 0 11 + 40x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.6875 + 2.5x + x2 = 0 Move the constant term to the right: Add '-0.6875' to each side of the equation. 0.6875 + 2.5x + -0.6875 + x2 = 0 + -0.6875 Reorder the terms: 0.6875 + -0.6875 + 2.5x + x2 = 0 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + 2.5x + x2 = 0 + -0.6875 2.5x + x2 = 0 + -0.6875 Combine like terms: 0 + -0.6875 = -0.6875 2.5x + x2 = -0.6875 The x term is 2.5x. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5x + 1.5625 + x2 = -0.6875 + 1.5625 Reorder the terms: 1.5625 + 2.5x + x2 = -0.6875 + 1.5625 Combine like terms: -0.6875 + 1.5625 = 0.875 1.5625 + 2.5x + x2 = 0.875 Factor a perfect square on the left side: (x + 1.25)(x + 1.25) = 0.875 Calculate the square root of the right side: 0.935414347 Break this problem into two subproblems by setting (x + 1.25) equal to 0.935414347 and -0.935414347.Subproblem 1
x + 1.25 = 0.935414347 Simplifying x + 1.25 = 0.935414347 Reorder the terms: 1.25 + x = 0.935414347 Solving 1.25 + x = 0.935414347 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = 0.935414347 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 0.935414347 + -1.25 x = 0.935414347 + -1.25 Combine like terms: 0.935414347 + -1.25 = -0.314585653 x = -0.314585653 Simplifying x = -0.314585653Subproblem 2
x + 1.25 = -0.935414347 Simplifying x + 1.25 = -0.935414347 Reorder the terms: 1.25 + x = -0.935414347 Solving 1.25 + x = -0.935414347 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = -0.935414347 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -0.935414347 + -1.25 x = -0.935414347 + -1.25 Combine like terms: -0.935414347 + -1.25 = -2.185414347 x = -2.185414347 Simplifying x = -2.185414347Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.314585653, -2.185414347}
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