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Simplifying -1(x + 1)(x + 1) + 8 = 0 Reorder the terms: -1(1 + x)(x + 1) + 8 = 0 Reorder the terms: -1(1 + x)(1 + x) + 8 = 0 Multiply (1 + x) * (1 + x) -1(1(1 + x) + x(1 + x)) + 8 = 0 -1((1 * 1 + x * 1) + x(1 + x)) + 8 = 0 -1((1 + 1x) + x(1 + x)) + 8 = 0 -1(1 + 1x + (1 * x + x * x)) + 8 = 0 -1(1 + 1x + (1x + x2)) + 8 = 0 Combine like terms: 1x + 1x = 2x -1(1 + 2x + x2) + 8 = 0 (1 * -1 + 2x * -1 + x2 * -1) + 8 = 0 (-1 + -2x + -1x2) + 8 = 0 Reorder the terms: -1 + 8 + -2x + -1x2 = 0 Combine like terms: -1 + 8 = 7 7 + -2x + -1x2 = 0 Solving 7 + -2x + -1x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -7 + 2x + x2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + 2x + 7 + x2 = 0 + 7 Reorder the terms: -7 + 7 + 2x + x2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 2x + x2 = 0 + 7 2x + x2 = 0 + 7 Combine like terms: 0 + 7 = 7 2x + x2 = 7 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 = 7 + 1 Reorder the terms: 1 + 2x + x2 = 7 + 1 Combine like terms: 7 + 1 = 8 1 + 2x + x2 = 8 Factor a perfect square on the left side: (x + 1)(x + 1) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (x + 1) equal to 2.828427125 and -2.828427125.Subproblem 1
x + 1 = 2.828427125 Simplifying x + 1 = 2.828427125 Reorder the terms: 1 + x = 2.828427125 Solving 1 + x = 2.828427125 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 = 2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + x = 2.828427125 + -1 x = 2.828427125 + -1 Combine like terms: 2.828427125 + -1 = 1.828427125 x = 1.828427125 Simplifying x = 1.828427125Subproblem 2
x + 1 = -2.828427125 Simplifying x + 1 = -2.828427125 Reorder the terms: 1 + x = -2.828427125 Solving 1 + x = -2.828427125 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 = -2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + x = -2.828427125 + -1 x = -2.828427125 + -1 Combine like terms: -2.828427125 + -1 = -3.828427125 x = -3.828427125 Simplifying x = -3.828427125Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.828427125, -3.828427125}
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