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Simplifying (x + -3)(2x + 7) = 1 Reorder the terms: (-3 + x)(2x + 7) = 1 Reorder the terms: (-3 + x)(7 + 2x) = 1 Multiply (-3 + x) * (7 + 2x) (-3(7 + 2x) + x(7 + 2x)) = 1 ((7 * -3 + 2x * -3) + x(7 + 2x)) = 1 ((-21 + -6x) + x(7 + 2x)) = 1 (-21 + -6x + (7 * x + 2x * x)) = 1 (-21 + -6x + (7x + 2x2)) = 1 Combine like terms: -6x + 7x = 1x (-21 + 1x + 2x2) = 1 Solving -21 + 1x + 2x2 = 1 Solving for variable 'x'. Reorder the terms: -21 + -1 + 1x + 2x2 = 1 + -1 Combine like terms: -21 + -1 = -22 -22 + 1x + 2x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -22 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -11 + 0.5x + x2 = 0 Move the constant term to the right: Add '11' to each side of the equation. -11 + 0.5x + 11 + x2 = 0 + 11 Reorder the terms: -11 + 11 + 0.5x + x2 = 0 + 11 Combine like terms: -11 + 11 = 0 0 + 0.5x + x2 = 0 + 11 0.5x + x2 = 0 + 11 Combine like terms: 0 + 11 = 11 0.5x + x2 = 11 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 11 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 11 + 0.0625 Combine like terms: 11 + 0.0625 = 11.0625 0.0625 + 0.5x + x2 = 11.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 11.0625 Calculate the square root of the right side: 3.326033674 Break this problem into two subproblems by setting (x + 0.25) equal to 3.326033674 and -3.326033674.Subproblem 1
x + 0.25 = 3.326033674 Simplifying x + 0.25 = 3.326033674 Reorder the terms: 0.25 + x = 3.326033674 Solving 0.25 + x = 3.326033674 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 3.326033674 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 3.326033674 + -0.25 x = 3.326033674 + -0.25 Combine like terms: 3.326033674 + -0.25 = 3.076033674 x = 3.076033674 Simplifying x = 3.076033674Subproblem 2
x + 0.25 = -3.326033674 Simplifying x + 0.25 = -3.326033674 Reorder the terms: 0.25 + x = -3.326033674 Solving 0.25 + x = -3.326033674 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -3.326033674 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -3.326033674 + -0.25 x = -3.326033674 + -0.25 Combine like terms: -3.326033674 + -0.25 = -3.576033674 x = -3.576033674 Simplifying x = -3.576033674Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.076033674, -3.576033674}
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