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Simplifying (7x + 5)(7x + 6) = 7 Reorder the terms: (5 + 7x)(7x + 6) = 7 Reorder the terms: (5 + 7x)(6 + 7x) = 7 Multiply (5 + 7x) * (6 + 7x) (5(6 + 7x) + 7x * (6 + 7x)) = 7 ((6 * 5 + 7x * 5) + 7x * (6 + 7x)) = 7 ((30 + 35x) + 7x * (6 + 7x)) = 7 (30 + 35x + (6 * 7x + 7x * 7x)) = 7 (30 + 35x + (42x + 49x2)) = 7 Combine like terms: 35x + 42x = 77x (30 + 77x + 49x2) = 7 Solving 30 + 77x + 49x2 = 7 Solving for variable 'x'. Reorder the terms: 30 + -7 + 77x + 49x2 = 7 + -7 Combine like terms: 30 + -7 = 23 23 + 77x + 49x2 = 7 + -7 Combine like terms: 7 + -7 = 0 23 + 77x + 49x2 = 0 Begin completing the square. Divide all terms by 49 the coefficient of the squared term: Divide each side by '49'. 0.4693877551 + 1.571428571x + x2 = 0 Move the constant term to the right: Add '-0.4693877551' to each side of the equation. 0.4693877551 + 1.571428571x + -0.4693877551 + x2 = 0 + -0.4693877551 Reorder the terms: 0.4693877551 + -0.4693877551 + 1.571428571x + x2 = 0 + -0.4693877551 Combine like terms: 0.4693877551 + -0.4693877551 = 0.0000000000 0.0000000000 + 1.571428571x + x2 = 0 + -0.4693877551 1.571428571x + x2 = 0 + -0.4693877551 Combine like terms: 0 + -0.4693877551 = -0.4693877551 1.571428571x + x2 = -0.4693877551 The x term is 1.571428571x. Take half its coefficient (0.7857142855). Square it (0.6173469384) and add it to both sides. Add '0.6173469384' to each side of the equation. 1.571428571x + 0.6173469384 + x2 = -0.4693877551 + 0.6173469384 Reorder the terms: 0.6173469384 + 1.571428571x + x2 = -0.4693877551 + 0.6173469384 Combine like terms: -0.4693877551 + 0.6173469384 = 0.1479591833 0.6173469384 + 1.571428571x + x2 = 0.1479591833 Factor a perfect square on the left side: (x + 0.7857142855)(x + 0.7857142855) = 0.1479591833 Calculate the square root of the right side: 0.384654629 Break this problem into two subproblems by setting (x + 0.7857142855) equal to 0.384654629 and -0.384654629.Subproblem 1
x + 0.7857142855 = 0.384654629 Simplifying x + 0.7857142855 = 0.384654629 Reorder the terms: 0.7857142855 + x = 0.384654629 Solving 0.7857142855 + x = 0.384654629 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + x = 0.384654629 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + x = 0.384654629 + -0.7857142855 x = 0.384654629 + -0.7857142855 Combine like terms: 0.384654629 + -0.7857142855 = -0.4010596565 x = -0.4010596565 Simplifying x = -0.4010596565Subproblem 2
x + 0.7857142855 = -0.384654629 Simplifying x + 0.7857142855 = -0.384654629 Reorder the terms: 0.7857142855 + x = -0.384654629 Solving 0.7857142855 + x = -0.384654629 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + x = -0.384654629 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + x = -0.384654629 + -0.7857142855 x = -0.384654629 + -0.7857142855 Combine like terms: -0.384654629 + -0.7857142855 = -1.1703689145 x = -1.1703689145 Simplifying x = -1.1703689145Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.4010596565, -1.1703689145}
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