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Simplifying 3x(x + 2(4x + 1) + 3) = 19 Reorder the terms: 3x(x + 2(1 + 4x) + 3) = 19 3x(x + (1 * 2 + 4x * 2) + 3) = 19 3x(x + (2 + 8x) + 3) = 19 Reorder the terms: 3x(2 + 3 + x + 8x) = 19 Combine like terms: 2 + 3 = 5 3x(5 + x + 8x) = 19 Combine like terms: x + 8x = 9x 3x(5 + 9x) = 19 (5 * 3x + 9x * 3x) = 19 (15x + 27x2) = 19 Solving 15x + 27x2 = 19 Solving for variable 'x'. Reorder the terms: -19 + 15x + 27x2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + 15x + 27x2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.7037037037 + 0.5555555556x + x2 = 0 Move the constant term to the right: Add '0.7037037037' to each side of the equation. -0.7037037037 + 0.5555555556x + 0.7037037037 + x2 = 0 + 0.7037037037 Reorder the terms: -0.7037037037 + 0.7037037037 + 0.5555555556x + x2 = 0 + 0.7037037037 Combine like terms: -0.7037037037 + 0.7037037037 = 0.0000000000 0.0000000000 + 0.5555555556x + x2 = 0 + 0.7037037037 0.5555555556x + x2 = 0 + 0.7037037037 Combine like terms: 0 + 0.7037037037 = 0.7037037037 0.5555555556x + x2 = 0.7037037037 The x term is 0.5555555556x. Take half its coefficient (0.2777777778). Square it (0.07716049384) and add it to both sides. Add '0.07716049384' to each side of the equation. 0.5555555556x + 0.07716049384 + x2 = 0.7037037037 + 0.07716049384 Reorder the terms: 0.07716049384 + 0.5555555556x + x2 = 0.7037037037 + 0.07716049384 Combine like terms: 0.7037037037 + 0.07716049384 = 0.78086419754 0.07716049384 + 0.5555555556x + x2 = 0.78086419754 Factor a perfect square on the left side: (x + 0.2777777778)(x + 0.2777777778) = 0.78086419754 Calculate the square root of the right side: 0.883665207 Break this problem into two subproblems by setting (x + 0.2777777778) equal to 0.883665207 and -0.883665207.Subproblem 1
x + 0.2777777778 = 0.883665207 Simplifying x + 0.2777777778 = 0.883665207 Reorder the terms: 0.2777777778 + x = 0.883665207 Solving 0.2777777778 + x = 0.883665207 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + x = 0.883665207 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + x = 0.883665207 + -0.2777777778 x = 0.883665207 + -0.2777777778 Combine like terms: 0.883665207 + -0.2777777778 = 0.6058874292 x = 0.6058874292 Simplifying x = 0.6058874292Subproblem 2
x + 0.2777777778 = -0.883665207 Simplifying x + 0.2777777778 = -0.883665207 Reorder the terms: 0.2777777778 + x = -0.883665207 Solving 0.2777777778 + x = -0.883665207 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + x = -0.883665207 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + x = -0.883665207 + -0.2777777778 x = -0.883665207 + -0.2777777778 Combine like terms: -0.883665207 + -0.2777777778 = -1.1614429848 x = -1.1614429848 Simplifying x = -1.1614429848Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.6058874292, -1.1614429848}
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