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Simplifying 3x(x + 2) = 5 Reorder the terms: 3x(2 + x) = 5 (2 * 3x + x * 3x) = 5 (6x + 3x2) = 5 Solving 6x + 3x2 = 5 Solving for variable 'x'. Reorder the terms: -5 + 6x + 3x2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + 6x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.666666667 + 2x + x2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 2x + 1.666666667 + x2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 2x + x2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 2x + x2 = 0 + 1.666666667 2x + x2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 2x + x2 = 1.666666667 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 = 1.666666667 + 1 Reorder the terms: 1 + 2x + x2 = 1.666666667 + 1 Combine like terms: 1.666666667 + 1 = 2.666666667 1 + 2x + x2 = 2.666666667 Factor a perfect square on the left side: (x + 1)(x + 1) = 2.666666667 Calculate the square root of the right side: 1.632993162 Break this problem into two subproblems by setting (x + 1) equal to 1.632993162 and -1.632993162.Subproblem 1
x + 1 = 1.632993162 Simplifying x + 1 = 1.632993162 Reorder the terms: 1 + x = 1.632993162 Solving 1 + x = 1.632993162 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 = 1.632993162 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.632993162 + -1 x = 1.632993162 + -1 Combine like terms: 1.632993162 + -1 = 0.632993162 x = 0.632993162 Simplifying x = 0.632993162Subproblem 2
x + 1 = -1.632993162 Simplifying x + 1 = -1.632993162 Reorder the terms: 1 + x = -1.632993162 Solving 1 + x = -1.632993162 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 = -1.632993162 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.632993162 + -1 x = -1.632993162 + -1 Combine like terms: -1.632993162 + -1 = -2.632993162 x = -2.632993162 Simplifying x = -2.632993162Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.632993162, -2.632993162}
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