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Simplifying 16x2 + 32x + -77 = 0 Reorder the terms: -77 + 32x + 16x2 = 0 Solving -77 + 32x + 16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -4.8125 + 2x + x2 = 0 Move the constant term to the right: Add '4.8125' to each side of the equation. -4.8125 + 2x + 4.8125 + x2 = 0 + 4.8125 Reorder the terms: -4.8125 + 4.8125 + 2x + x2 = 0 + 4.8125 Combine like terms: -4.8125 + 4.8125 = 0.0000 0.0000 + 2x + x2 = 0 + 4.8125 2x + x2 = 0 + 4.8125 Combine like terms: 0 + 4.8125 = 4.8125 2x + x2 = 4.8125 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 4.8125 + 1 Reorder the terms: 1 + 2x + x2 = 4.8125 + 1 Combine like terms: 4.8125 + 1 = 5.8125 1 + 2x + x2 = 5.8125 Factor a perfect square on the left side: (x + 1)(x + 1) = 5.8125 Calculate the square root of the right side: 2.41091269 Break this problem into two subproblems by setting (x + 1) equal to 2.41091269 and -2.41091269.Subproblem 1
x + 1 = 2.41091269 Simplifying x + 1 = 2.41091269 Reorder the terms: 1 + x = 2.41091269 Solving 1 + x = 2.41091269 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 2.41091269 + -1 Combine like terms: 1 + -1 = 0 0 + x = 2.41091269 + -1 x = 2.41091269 + -1 Combine like terms: 2.41091269 + -1 = 1.41091269 x = 1.41091269 Simplifying x = 1.41091269Subproblem 2
x + 1 = -2.41091269 Simplifying x + 1 = -2.41091269 Reorder the terms: 1 + x = -2.41091269 Solving 1 + x = -2.41091269 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -2.41091269 + -1 Combine like terms: 1 + -1 = 0 0 + x = -2.41091269 + -1 x = -2.41091269 + -1 Combine like terms: -2.41091269 + -1 = -3.41091269 x = -3.41091269 Simplifying x = -3.41091269Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.41091269, -3.41091269}
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