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Simplifying (4x + 3)(x + 1) = 1 Reorder the terms: (3 + 4x)(x + 1) = 1 Reorder the terms: (3 + 4x)(1 + x) = 1 Multiply (3 + 4x) * (1 + x) (3(1 + x) + 4x * (1 + x)) = 1 ((1 * 3 + x * 3) + 4x * (1 + x)) = 1 ((3 + 3x) + 4x * (1 + x)) = 1 (3 + 3x + (1 * 4x + x * 4x)) = 1 (3 + 3x + (4x + 4x2)) = 1 Combine like terms: 3x + 4x = 7x (3 + 7x + 4x2) = 1 Solving 3 + 7x + 4x2 = 1 Solving for variable 'x'. Reorder the terms: 3 + -1 + 7x + 4x2 = 1 + -1 Combine like terms: 3 + -1 = 2 2 + 7x + 4x2 = 1 + -1 Combine like terms: 1 + -1 = 0 2 + 7x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.5 + 1.75x + x2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 1.75x + -0.5 + x2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 1.75x + x2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 1.75x + x2 = 0 + -0.5 1.75x + x2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 1.75x + x2 = -0.5 The x term is 1.75x. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75x + 0.765625 + x2 = -0.5 + 0.765625 Reorder the terms: 0.765625 + 1.75x + x2 = -0.5 + 0.765625 Combine like terms: -0.5 + 0.765625 = 0.265625 0.765625 + 1.75x + x2 = 0.265625 Factor a perfect square on the left side: (x + 0.875)(x + 0.875) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (x + 0.875) equal to 0.515388203 and -0.515388203.Subproblem 1
x + 0.875 = 0.515388203 Simplifying x + 0.875 = 0.515388203 Reorder the terms: 0.875 + x = 0.515388203 Solving 0.875 + x = 0.515388203 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = 0.515388203 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = 0.515388203 + -0.875 x = 0.515388203 + -0.875 Combine like terms: 0.515388203 + -0.875 = -0.359611797 x = -0.359611797 Simplifying x = -0.359611797Subproblem 2
x + 0.875 = -0.515388203 Simplifying x + 0.875 = -0.515388203 Reorder the terms: 0.875 + x = -0.515388203 Solving 0.875 + x = -0.515388203 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = -0.515388203 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = -0.515388203 + -0.875 x = -0.515388203 + -0.875 Combine like terms: -0.515388203 + -0.875 = -1.390388203 x = -1.390388203 Simplifying x = -1.390388203Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.359611797, -1.390388203}
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