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Simplifying x2 + 2x = 16384 Reorder the terms: 2x + x2 = 16384 Solving 2x + x2 = 16384 Solving for variable 'x'. Reorder the terms: -16384 + 2x + x2 = 16384 + -16384 Combine like terms: 16384 + -16384 = 0 -16384 + 2x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '16384' to each side of the equation. -16384 + 2x + 16384 + x2 = 0 + 16384 Reorder the terms: -16384 + 16384 + 2x + x2 = 0 + 16384 Combine like terms: -16384 + 16384 = 0 0 + 2x + x2 = 0 + 16384 2x + x2 = 0 + 16384 Combine like terms: 0 + 16384 = 16384 2x + x2 = 16384 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 = 16384 + 1 Reorder the terms: 1 + 2x + x2 = 16384 + 1 Combine like terms: 16384 + 1 = 16385 1 + 2x + x2 = 16385 Factor a perfect square on the left side: (x + 1)(x + 1) = 16385 Calculate the square root of the right side: 128.00390619 Break this problem into two subproblems by setting (x + 1) equal to 128.00390619 and -128.00390619.Subproblem 1
x + 1 = 128.00390619 Simplifying x + 1 = 128.00390619 Reorder the terms: 1 + x = 128.00390619 Solving 1 + x = 128.00390619 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 = 128.00390619 + -1 Combine like terms: 1 + -1 = 0 0 + x = 128.00390619 + -1 x = 128.00390619 + -1 Combine like terms: 128.00390619 + -1 = 127.00390619 x = 127.00390619 Simplifying x = 127.00390619Subproblem 2
x + 1 = -128.00390619 Simplifying x + 1 = -128.00390619 Reorder the terms: 1 + x = -128.00390619 Solving 1 + x = -128.00390619 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 = -128.00390619 + -1 Combine like terms: 1 + -1 = 0 0 + x = -128.00390619 + -1 x = -128.00390619 + -1 Combine like terms: -128.00390619 + -1 = -129.00390619 x = -129.00390619 Simplifying x = -129.00390619Solution
The solution to the problem is based on the solutions from the subproblems. x = {127.00390619, -129.00390619}
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