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Simplifying 8x(x + 2) + 5 = 5x(x + 1) Reorder the terms: 8x(2 + x) + 5 = 5x(x + 1) (2 * 8x + x * 8x) + 5 = 5x(x + 1) (16x + 8x2) + 5 = 5x(x + 1) Reorder the terms: 5 + 16x + 8x2 = 5x(x + 1) Reorder the terms: 5 + 16x + 8x2 = 5x(1 + x) 5 + 16x + 8x2 = (1 * 5x + x * 5x) 5 + 16x + 8x2 = (5x + 5x2) Solving 5 + 16x + 8x2 = 5x + 5x2 Solving for variable 'x'. Reorder the terms: 5 + 16x + -5x + 8x2 + -5x2 = 5x + 5x2 + -5x + -5x2 Combine like terms: 16x + -5x = 11x 5 + 11x + 8x2 + -5x2 = 5x + 5x2 + -5x + -5x2 Combine like terms: 8x2 + -5x2 = 3x2 5 + 11x + 3x2 = 5x + 5x2 + -5x + -5x2 Reorder the terms: 5 + 11x + 3x2 = 5x + -5x + 5x2 + -5x2 Combine like terms: 5x + -5x = 0 5 + 11x + 3x2 = 0 + 5x2 + -5x2 5 + 11x + 3x2 = 5x2 + -5x2 Combine like terms: 5x2 + -5x2 = 0 5 + 11x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.666666667 + 3.666666667x + x2 = 0 Move the constant term to the right: Add '-1.666666667' to each side of the equation. 1.666666667 + 3.666666667x + -1.666666667 + x2 = 0 + -1.666666667 Reorder the terms: 1.666666667 + -1.666666667 + 3.666666667x + x2 = 0 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + 3.666666667x + x2 = 0 + -1.666666667 3.666666667x + x2 = 0 + -1.666666667 Combine like terms: 0 + -1.666666667 = -1.666666667 3.666666667x + x2 = -1.666666667 The x term is 3.666666667x. Take half its coefficient (1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. 3.666666667x + 3.361111114 + x2 = -1.666666667 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667x + x2 = -1.666666667 + 3.361111114 Combine like terms: -1.666666667 + 3.361111114 = 1.694444447 3.361111114 + 3.666666667x + x2 = 1.694444447 Factor a perfect square on the left side: (x + 1.833333334)(x + 1.833333334) = 1.694444447 Calculate the square root of the right side: 1.30170828 Break this problem into two subproblems by setting (x + 1.833333334) equal to 1.30170828 and -1.30170828.Subproblem 1
x + 1.833333334 = 1.30170828 Simplifying x + 1.833333334 = 1.30170828 Reorder the terms: 1.833333334 + x = 1.30170828 Solving 1.833333334 + x = 1.30170828 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = 1.30170828 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = 1.30170828 + -1.833333334 x = 1.30170828 + -1.833333334 Combine like terms: 1.30170828 + -1.833333334 = -0.531625054 x = -0.531625054 Simplifying x = -0.531625054Subproblem 2
x + 1.833333334 = -1.30170828 Simplifying x + 1.833333334 = -1.30170828 Reorder the terms: 1.833333334 + x = -1.30170828 Solving 1.833333334 + x = -1.30170828 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = -1.30170828 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = -1.30170828 + -1.833333334 x = -1.30170828 + -1.833333334 Combine like terms: -1.30170828 + -1.833333334 = -3.135041614 x = -3.135041614 Simplifying x = -3.135041614Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.531625054, -3.135041614}
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