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Simplifying 3x(4 + x) + 5(x + -3) = 38 (4 * 3x + x * 3x) + 5(x + -3) = 38 (12x + 3x2) + 5(x + -3) = 38 Reorder the terms: 12x + 3x2 + 5(-3 + x) = 38 12x + 3x2 + (-3 * 5 + x * 5) = 38 12x + 3x2 + (-15 + 5x) = 38 Reorder the terms: -15 + 12x + 5x + 3x2 = 38 Combine like terms: 12x + 5x = 17x -15 + 17x + 3x2 = 38 Solving -15 + 17x + 3x2 = 38 Solving for variable 'x'. Reorder the terms: -15 + -38 + 17x + 3x2 = 38 + -38 Combine like terms: -15 + -38 = -53 -53 + 17x + 3x2 = 38 + -38 Combine like terms: 38 + -38 = 0 -53 + 17x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -17.66666667 + 5.666666667x + x2 = 0 Move the constant term to the right: Add '17.66666667' to each side of the equation. -17.66666667 + 5.666666667x + 17.66666667 + x2 = 0 + 17.66666667 Reorder the terms: -17.66666667 + 17.66666667 + 5.666666667x + x2 = 0 + 17.66666667 Combine like terms: -17.66666667 + 17.66666667 = 0.00000000 0.00000000 + 5.666666667x + x2 = 0 + 17.66666667 5.666666667x + x2 = 0 + 17.66666667 Combine like terms: 0 + 17.66666667 = 17.66666667 5.666666667x + x2 = 17.66666667 The x term is 5.666666667x. Take half its coefficient (2.833333334). Square it (8.027777782) and add it to both sides. Add '8.027777782' to each side of the equation. 5.666666667x + 8.027777782 + x2 = 17.66666667 + 8.027777782 Reorder the terms: 8.027777782 + 5.666666667x + x2 = 17.66666667 + 8.027777782 Combine like terms: 17.66666667 + 8.027777782 = 25.694444452 8.027777782 + 5.666666667x + x2 = 25.694444452 Factor a perfect square on the left side: (x + 2.833333334)(x + 2.833333334) = 25.694444452 Calculate the square root of the right side: 5.068968776 Break this problem into two subproblems by setting (x + 2.833333334) equal to 5.068968776 and -5.068968776.Subproblem 1
x + 2.833333334 = 5.068968776 Simplifying x + 2.833333334 = 5.068968776 Reorder the terms: 2.833333334 + x = 5.068968776 Solving 2.833333334 + x = 5.068968776 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.833333334' to each side of the equation. 2.833333334 + -2.833333334 + x = 5.068968776 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + x = 5.068968776 + -2.833333334 x = 5.068968776 + -2.833333334 Combine like terms: 5.068968776 + -2.833333334 = 2.235635442 x = 2.235635442 Simplifying x = 2.235635442Subproblem 2
x + 2.833333334 = -5.068968776 Simplifying x + 2.833333334 = -5.068968776 Reorder the terms: 2.833333334 + x = -5.068968776 Solving 2.833333334 + x = -5.068968776 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.833333334' to each side of the equation. 2.833333334 + -2.833333334 + x = -5.068968776 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + x = -5.068968776 + -2.833333334 x = -5.068968776 + -2.833333334 Combine like terms: -5.068968776 + -2.833333334 = -7.90230211 x = -7.90230211 Simplifying x = -7.90230211Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.235635442, -7.90230211}
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