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Simplifying (4x + 14)(7x + -2) = 1 Reorder the terms: (14 + 4x)(7x + -2) = 1 Reorder the terms: (14 + 4x)(-2 + 7x) = 1 Multiply (14 + 4x) * (-2 + 7x) (14(-2 + 7x) + 4x * (-2 + 7x)) = 1 ((-2 * 14 + 7x * 14) + 4x * (-2 + 7x)) = 1 ((-28 + 98x) + 4x * (-2 + 7x)) = 1 (-28 + 98x + (-2 * 4x + 7x * 4x)) = 1 (-28 + 98x + (-8x + 28x2)) = 1 Combine like terms: 98x + -8x = 90x (-28 + 90x + 28x2) = 1 Solving -28 + 90x + 28x2 = 1 Solving for variable 'x'. Reorder the terms: -28 + -1 + 90x + 28x2 = 1 + -1 Combine like terms: -28 + -1 = -29 -29 + 90x + 28x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -29 + 90x + 28x2 = 0 Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. -1.035714286 + 3.214285714x + x2 = 0 Move the constant term to the right: Add '1.035714286' to each side of the equation. -1.035714286 + 3.214285714x + 1.035714286 + x2 = 0 + 1.035714286 Reorder the terms: -1.035714286 + 1.035714286 + 3.214285714x + x2 = 0 + 1.035714286 Combine like terms: -1.035714286 + 1.035714286 = 0.000000000 0.000000000 + 3.214285714x + x2 = 0 + 1.035714286 3.214285714x + x2 = 0 + 1.035714286 Combine like terms: 0 + 1.035714286 = 1.035714286 3.214285714x + x2 = 1.035714286 The x term is 3.214285714x. Take half its coefficient (1.607142857). Square it (2.582908163) and add it to both sides. Add '2.582908163' to each side of the equation. 3.214285714x + 2.582908163 + x2 = 1.035714286 + 2.582908163 Reorder the terms: 2.582908163 + 3.214285714x + x2 = 1.035714286 + 2.582908163 Combine like terms: 1.035714286 + 2.582908163 = 3.618622449 2.582908163 + 3.214285714x + x2 = 3.618622449 Factor a perfect square on the left side: (x + 1.607142857)(x + 1.607142857) = 3.618622449 Calculate the square root of the right side: 1.902267712 Break this problem into two subproblems by setting (x + 1.607142857) equal to 1.902267712 and -1.902267712.Subproblem 1
x + 1.607142857 = 1.902267712 Simplifying x + 1.607142857 = 1.902267712 Reorder the terms: 1.607142857 + x = 1.902267712 Solving 1.607142857 + x = 1.902267712 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.607142857' to each side of the equation. 1.607142857 + -1.607142857 + x = 1.902267712 + -1.607142857 Combine like terms: 1.607142857 + -1.607142857 = 0.000000000 0.000000000 + x = 1.902267712 + -1.607142857 x = 1.902267712 + -1.607142857 Combine like terms: 1.902267712 + -1.607142857 = 0.295124855 x = 0.295124855 Simplifying x = 0.295124855Subproblem 2
x + 1.607142857 = -1.902267712 Simplifying x + 1.607142857 = -1.902267712 Reorder the terms: 1.607142857 + x = -1.902267712 Solving 1.607142857 + x = -1.902267712 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.607142857' to each side of the equation. 1.607142857 + -1.607142857 + x = -1.902267712 + -1.607142857 Combine like terms: 1.607142857 + -1.607142857 = 0.000000000 0.000000000 + x = -1.902267712 + -1.607142857 x = -1.902267712 + -1.607142857 Combine like terms: -1.902267712 + -1.607142857 = -3.509410569 x = -3.509410569 Simplifying x = -3.509410569Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.295124855, -3.509410569}
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