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Simplifying 8 = (x + 0.5)(x + 1) Reorder the terms: 8 = (0.5 + x)(x + 1) Reorder the terms: 8 = (0.5 + x)(1 + x) Multiply (0.5 + x) * (1 + x) 8 = (0.5(1 + x) + x(1 + x)) 8 = ((1 * 0.5 + x * 0.5) + x(1 + x)) 8 = ((0.5 + 0.5x) + x(1 + x)) 8 = (0.5 + 0.5x + (1 * x + x * x)) 8 = (0.5 + 0.5x + (1x + x2)) Combine like terms: 0.5x + 1x = 1.5x 8 = (0.5 + 1.5x + x2) Solving 8 = 0.5 + 1.5x + x2 Solving for variable 'x'. Combine like terms: 8 + -0.5 = 7.5 7.5 + -1.5x + -1x2 = 0.5 + 1.5x + x2 + -0.5 + -1.5x + -1x2 Reorder the terms: 7.5 + -1.5x + -1x2 = 0.5 + -0.5 + 1.5x + -1.5x + x2 + -1x2 Combine like terms: 0.5 + -0.5 = 0.0 7.5 + -1.5x + -1x2 = 0.0 + 1.5x + -1.5x + x2 + -1x2 7.5 + -1.5x + -1x2 = 1.5x + -1.5x + x2 + -1x2 Combine like terms: 1.5x + -1.5x = 0.0 7.5 + -1.5x + -1x2 = 0.0 + x2 + -1x2 7.5 + -1.5x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 7.5 + -1.5x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -7.5 + 1.5x + x2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 1.5x + 7.5 + x2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 1.5x + x2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 1.5x + x2 = 0 + 7.5 1.5x + x2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 1.5x + x2 = 7.5 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 = 7.5 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 7.5 + 0.5625 Combine like terms: 7.5 + 0.5625 = 8.0625 0.5625 + 1.5x + x2 = 8.0625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 8.0625 Calculate the square root of the right side: 2.839454173 Break this problem into two subproblems by setting (x + 0.75) equal to 2.839454173 and -2.839454173.Subproblem 1
x + 0.75 = 2.839454173 Simplifying x + 0.75 = 2.839454173 Reorder the terms: 0.75 + x = 2.839454173 Solving 0.75 + x = 2.839454173 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 = 2.839454173 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 2.839454173 + -0.75 x = 2.839454173 + -0.75 Combine like terms: 2.839454173 + -0.75 = 2.089454173 x = 2.089454173 Simplifying x = 2.089454173Subproblem 2
x + 0.75 = -2.839454173 Simplifying x + 0.75 = -2.839454173 Reorder the terms: 0.75 + x = -2.839454173 Solving 0.75 + x = -2.839454173 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 = -2.839454173 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -2.839454173 + -0.75 x = -2.839454173 + -0.75 Combine like terms: -2.839454173 + -0.75 = -3.589454173 x = -3.589454173 Simplifying x = -3.589454173Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.089454173, -3.589454173}
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