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Simplifying (3x + 14)(x + 1) = 10 Reorder the terms: (14 + 3x)(x + 1) = 10 Reorder the terms: (14 + 3x)(1 + x) = 10 Multiply (14 + 3x) * (1 + x) (14(1 + x) + 3x * (1 + x)) = 10 ((1 * 14 + x * 14) + 3x * (1 + x)) = 10 ((14 + 14x) + 3x * (1 + x)) = 10 (14 + 14x + (1 * 3x + x * 3x)) = 10 (14 + 14x + (3x + 3x2)) = 10 Combine like terms: 14x + 3x = 17x (14 + 17x + 3x2) = 10 Solving 14 + 17x + 3x2 = 10 Solving for variable 'x'. Reorder the terms: 14 + -10 + 17x + 3x2 = 10 + -10 Combine like terms: 14 + -10 = 4 4 + 17x + 3x2 = 10 + -10 Combine like terms: 10 + -10 = 0 4 + 17x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + 5.666666667x + x2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + 5.666666667x + -1.333333333 + x2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + 5.666666667x + x2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + 5.666666667x + x2 = 0 + -1.333333333 5.666666667x + x2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 5.666666667x + x2 = -1.333333333 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 = -1.333333333 + 8.027777782 Reorder the terms: 8.027777782 + 5.666666667x + x2 = -1.333333333 + 8.027777782 Combine like terms: -1.333333333 + 8.027777782 = 6.694444449 8.027777782 + 5.666666667x + x2 = 6.694444449 Factor a perfect square on the left side: (x + 2.833333334)(x + 2.833333334) = 6.694444449 Calculate the square root of the right side: 2.58736245 Break this problem into two subproblems by setting (x + 2.833333334) equal to 2.58736245 and -2.58736245.Subproblem 1
x + 2.833333334 = 2.58736245 Simplifying x + 2.833333334 = 2.58736245 Reorder the terms: 2.833333334 + x = 2.58736245 Solving 2.833333334 + x = 2.58736245 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 = 2.58736245 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + x = 2.58736245 + -2.833333334 x = 2.58736245 + -2.833333334 Combine like terms: 2.58736245 + -2.833333334 = -0.245970884 x = -0.245970884 Simplifying x = -0.245970884Subproblem 2
x + 2.833333334 = -2.58736245 Simplifying x + 2.833333334 = -2.58736245 Reorder the terms: 2.833333334 + x = -2.58736245 Solving 2.833333334 + x = -2.58736245 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 = -2.58736245 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + x = -2.58736245 + -2.833333334 x = -2.58736245 + -2.833333334 Combine like terms: -2.58736245 + -2.833333334 = -5.420695784 x = -5.420695784 Simplifying x = -5.420695784Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.245970884, -5.420695784}
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