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Simplifying 64 = (3x + 1)(x + 1) Reorder the terms: 64 = (1 + 3x)(x + 1) Reorder the terms: 64 = (1 + 3x)(1 + x) Multiply (1 + 3x) * (1 + x) 64 = (1(1 + x) + 3x * (1 + x)) 64 = ((1 * 1 + x * 1) + 3x * (1 + x)) 64 = ((1 + 1x) + 3x * (1 + x)) 64 = (1 + 1x + (1 * 3x + x * 3x)) 64 = (1 + 1x + (3x + 3x2)) Combine like terms: 1x + 3x = 4x 64 = (1 + 4x + 3x2) Solving 64 = 1 + 4x + 3x2 Solving for variable 'x'. Combine like terms: 64 + -1 = 63 63 + -4x + -3x2 = 1 + 4x + 3x2 + -1 + -4x + -3x2 Reorder the terms: 63 + -4x + -3x2 = 1 + -1 + 4x + -4x + 3x2 + -3x2 Combine like terms: 1 + -1 = 0 63 + -4x + -3x2 = 0 + 4x + -4x + 3x2 + -3x2 63 + -4x + -3x2 = 4x + -4x + 3x2 + -3x2 Combine like terms: 4x + -4x = 0 63 + -4x + -3x2 = 0 + 3x2 + -3x2 63 + -4x + -3x2 = 3x2 + -3x2 Combine like terms: 3x2 + -3x2 = 0 63 + -4x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -21 + 1.333333333x + x2 = 0 Move the constant term to the right: Add '21' to each side of the equation. -21 + 1.333333333x + 21 + x2 = 0 + 21 Reorder the terms: -21 + 21 + 1.333333333x + x2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + 1.333333333x + x2 = 0 + 21 1.333333333x + x2 = 0 + 21 Combine like terms: 0 + 21 = 21 1.333333333x + x2 = 21 The x term is 1.333333333x. Take half its coefficient (0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. 1.333333333x + 0.4444444442 + x2 = 21 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333x + x2 = 21 + 0.4444444442 Combine like terms: 21 + 0.4444444442 = 21.4444444442 0.4444444442 + 1.333333333x + x2 = 21.4444444442 Factor a perfect square on the left side: (x + 0.6666666665)(x + 0.6666666665) = 21.4444444442 Calculate the square root of the right side: 4.630814663 Break this problem into two subproblems by setting (x + 0.6666666665) equal to 4.630814663 and -4.630814663.Subproblem 1
x + 0.6666666665 = 4.630814663 Simplifying x + 0.6666666665 = 4.630814663 Reorder the terms: 0.6666666665 + x = 4.630814663 Solving 0.6666666665 + x = 4.630814663 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = 4.630814663 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 4.630814663 + -0.6666666665 x = 4.630814663 + -0.6666666665 Combine like terms: 4.630814663 + -0.6666666665 = 3.9641479965 x = 3.9641479965 Simplifying x = 3.9641479965Subproblem 2
x + 0.6666666665 = -4.630814663 Simplifying x + 0.6666666665 = -4.630814663 Reorder the terms: 0.6666666665 + x = -4.630814663 Solving 0.6666666665 + x = -4.630814663 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = -4.630814663 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -4.630814663 + -0.6666666665 x = -4.630814663 + -0.6666666665 Combine like terms: -4.630814663 + -0.6666666665 = -5.2974813295 x = -5.2974813295 Simplifying x = -5.2974813295Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.9641479965, -5.2974813295}
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