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Simplifying 3x(3x + 4) = 14 Reorder the terms: 3x(4 + 3x) = 14 (4 * 3x + 3x * 3x) = 14 (12x + 9x2) = 14 Solving 12x + 9x2 = 14 Solving for variable 'x'. Reorder the terms: -14 + 12x + 9x2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + 12x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.555555556 + 1.333333333x + x2 = 0 Move the constant term to the right: Add '1.555555556' to each side of the equation. -1.555555556 + 1.333333333x + 1.555555556 + x2 = 0 + 1.555555556 Reorder the terms: -1.555555556 + 1.555555556 + 1.333333333x + x2 = 0 + 1.555555556 Combine like terms: -1.555555556 + 1.555555556 = 0.000000000 0.000000000 + 1.333333333x + x2 = 0 + 1.555555556 1.333333333x + x2 = 0 + 1.555555556 Combine like terms: 0 + 1.555555556 = 1.555555556 1.333333333x + x2 = 1.555555556 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 = 1.555555556 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333x + x2 = 1.555555556 + 0.4444444442 Combine like terms: 1.555555556 + 0.4444444442 = 2.0000000002 0.4444444442 + 1.333333333x + x2 = 2.0000000002 Factor a perfect square on the left side: (x + 0.6666666665)(x + 0.6666666665) = 2.0000000002 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (x + 0.6666666665) equal to 1.414213562 and -1.414213562.Subproblem 1
x + 0.6666666665 = 1.414213562 Simplifying x + 0.6666666665 = 1.414213562 Reorder the terms: 0.6666666665 + x = 1.414213562 Solving 0.6666666665 + x = 1.414213562 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 = 1.414213562 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 1.414213562 + -0.6666666665 x = 1.414213562 + -0.6666666665 Combine like terms: 1.414213562 + -0.6666666665 = 0.7475468955 x = 0.7475468955 Simplifying x = 0.7475468955Subproblem 2
x + 0.6666666665 = -1.414213562 Simplifying x + 0.6666666665 = -1.414213562 Reorder the terms: 0.6666666665 + x = -1.414213562 Solving 0.6666666665 + x = -1.414213562 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 = -1.414213562 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -1.414213562 + -0.6666666665 x = -1.414213562 + -0.6666666665 Combine like terms: -1.414213562 + -0.6666666665 = -2.0808802285 x = -2.0808802285 Simplifying x = -2.0808802285Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7475468955, -2.0808802285}
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