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Simplifying (10x + 9)(7x + 1) = 38 Reorder the terms: (9 + 10x)(7x + 1) = 38 Reorder the terms: (9 + 10x)(1 + 7x) = 38 Multiply (9 + 10x) * (1 + 7x) (9(1 + 7x) + 10x * (1 + 7x)) = 38 ((1 * 9 + 7x * 9) + 10x * (1 + 7x)) = 38 ((9 + 63x) + 10x * (1 + 7x)) = 38 (9 + 63x + (1 * 10x + 7x * 10x)) = 38 (9 + 63x + (10x + 70x2)) = 38 Combine like terms: 63x + 10x = 73x (9 + 73x + 70x2) = 38 Solving 9 + 73x + 70x2 = 38 Solving for variable 'x'. Reorder the terms: 9 + -38 + 73x + 70x2 = 38 + -38 Combine like terms: 9 + -38 = -29 -29 + 73x + 70x2 = 38 + -38 Combine like terms: 38 + -38 = 0 -29 + 73x + 70x2 = 0 Begin completing the square. Divide all terms by 70 the coefficient of the squared term: Divide each side by '70'. -0.4142857143 + 1.042857143x + x2 = 0.0 Move the constant term to the right: Add '0.4142857143' to each side of the equation. -0.4142857143 + 1.042857143x + 0.4142857143 + x2 = 0.0 + 0.4142857143 Reorder the terms: -0.4142857143 + 0.4142857143 + 1.042857143x + x2 = 0.0 + 0.4142857143 Combine like terms: -0.4142857143 + 0.4142857143 = 0.0000000000 0.0000000000 + 1.042857143x + x2 = 0.0 + 0.4142857143 1.042857143x + x2 = 0.0 + 0.4142857143 Combine like terms: 0.0 + 0.4142857143 = 0.4142857143 1.042857143x + x2 = 0.4142857143 The x term is 1.042857143x. Take half its coefficient (0.5214285715). Square it (0.2718877552) and add it to both sides. Add '0.2718877552' to each side of the equation. 1.042857143x + 0.2718877552 + x2 = 0.4142857143 + 0.2718877552 Reorder the terms: 0.2718877552 + 1.042857143x + x2 = 0.4142857143 + 0.2718877552 Combine like terms: 0.4142857143 + 0.2718877552 = 0.6861734695 0.2718877552 + 1.042857143x + x2 = 0.6861734695 Factor a perfect square on the left side: (x + 0.5214285715)(x + 0.5214285715) = 0.6861734695 Calculate the square root of the right side: 0.828355883 Break this problem into two subproblems by setting (x + 0.5214285715) equal to 0.828355883 and -0.828355883.Subproblem 1
x + 0.5214285715 = 0.828355883 Simplifying x + 0.5214285715 = 0.828355883 Reorder the terms: 0.5214285715 + x = 0.828355883 Solving 0.5214285715 + x = 0.828355883 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5214285715' to each side of the equation. 0.5214285715 + -0.5214285715 + x = 0.828355883 + -0.5214285715 Combine like terms: 0.5214285715 + -0.5214285715 = 0.0000000000 0.0000000000 + x = 0.828355883 + -0.5214285715 x = 0.828355883 + -0.5214285715 Combine like terms: 0.828355883 + -0.5214285715 = 0.3069273115 x = 0.3069273115 Simplifying x = 0.3069273115Subproblem 2
x + 0.5214285715 = -0.828355883 Simplifying x + 0.5214285715 = -0.828355883 Reorder the terms: 0.5214285715 + x = -0.828355883 Solving 0.5214285715 + x = -0.828355883 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5214285715' to each side of the equation. 0.5214285715 + -0.5214285715 + x = -0.828355883 + -0.5214285715 Combine like terms: 0.5214285715 + -0.5214285715 = 0.0000000000 0.0000000000 + x = -0.828355883 + -0.5214285715 x = -0.828355883 + -0.5214285715 Combine like terms: -0.828355883 + -0.5214285715 = -1.3497844545 x = -1.3497844545 Simplifying x = -1.3497844545Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.3069273115, -1.3497844545}
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