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Simplifying (3x + 7)(x + 1) = 10 Reorder the terms: (7 + 3x)(x + 1) = 10 Reorder the terms: (7 + 3x)(1 + x) = 10 Multiply (7 + 3x) * (1 + x) (7(1 + x) + 3x * (1 + x)) = 10 ((1 * 7 + x * 7) + 3x * (1 + x)) = 10 ((7 + 7x) + 3x * (1 + x)) = 10 (7 + 7x + (1 * 3x + x * 3x)) = 10 (7 + 7x + (3x + 3x2)) = 10 Combine like terms: 7x + 3x = 10x (7 + 10x + 3x2) = 10 Solving 7 + 10x + 3x2 = 10 Solving for variable 'x'. Reorder the terms: 7 + -10 + 10x + 3x2 = 10 + -10 Combine like terms: 7 + -10 = -3 -3 + 10x + 3x2 = 10 + -10 Combine like terms: 10 + -10 = 0 -3 + 10x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1 + 3.333333333x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 3.333333333x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + 3.333333333x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 3.333333333x + x2 = 0 + 1 3.333333333x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 3.333333333x + x2 = 1 The x term is 3.333333333x. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333x + 2.777777779 + x2 = 1 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333x + x2 = 1 + 2.777777779 Combine like terms: 1 + 2.777777779 = 3.777777779 2.777777779 + 3.333333333x + x2 = 3.777777779 Factor a perfect square on the left side: (x + 1.666666667)(x + 1.666666667) = 3.777777779 Calculate the square root of the right side: 1.943650632 Break this problem into two subproblems by setting (x + 1.666666667) equal to 1.943650632 and -1.943650632.Subproblem 1
x + 1.666666667 = 1.943650632 Simplifying x + 1.666666667 = 1.943650632 Reorder the terms: 1.666666667 + x = 1.943650632 Solving 1.666666667 + x = 1.943650632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = 1.943650632 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = 1.943650632 + -1.666666667 x = 1.943650632 + -1.666666667 Combine like terms: 1.943650632 + -1.666666667 = 0.276983965 x = 0.276983965 Simplifying x = 0.276983965Subproblem 2
x + 1.666666667 = -1.943650632 Simplifying x + 1.666666667 = -1.943650632 Reorder the terms: 1.666666667 + x = -1.943650632 Solving 1.666666667 + x = -1.943650632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = -1.943650632 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = -1.943650632 + -1.666666667 x = -1.943650632 + -1.666666667 Combine like terms: -1.943650632 + -1.666666667 = -3.610317299 x = -3.610317299 Simplifying x = -3.610317299Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.276983965, -3.610317299}
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