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Simplifying 7x + -11y + -23 = x(x + 3) + -11(y + x) Reorder the terms: -23 + 7x + -11y = x(x + 3) + -11(y + x) Reorder the terms: -23 + 7x + -11y = x(3 + x) + -11(y + x) -23 + 7x + -11y = (3 * x + x * x) + -11(y + x) -23 + 7x + -11y = (3x + x2) + -11(y + x) Reorder the terms: -23 + 7x + -11y = 3x + x2 + -11(x + y) -23 + 7x + -11y = 3x + x2 + (x * -11 + y * -11) -23 + 7x + -11y = 3x + x2 + (-11x + -11y) Reorder the terms: -23 + 7x + -11y = 3x + -11x + x2 + -11y Combine like terms: 3x + -11x = -8x -23 + 7x + -11y = -8x + x2 + -11y Add '11y' to each side of the equation. -23 + 7x + -11y + 11y = -8x + x2 + -11y + 11y Combine like terms: -11y + 11y = 0 -23 + 7x + 0 = -8x + x2 + -11y + 11y -23 + 7x = -8x + x2 + -11y + 11y Combine like terms: -11y + 11y = 0 -23 + 7x = -8x + x2 + 0 -23 + 7x = -8x + x2 Solving -23 + 7x = -8x + x2 Solving for variable 'x'. Combine like terms: 7x + 8x = 15x -23 + 15x + -1x2 = -8x + x2 + 8x + -1x2 Reorder the terms: -23 + 15x + -1x2 = -8x + 8x + x2 + -1x2 Combine like terms: -8x + 8x = 0 -23 + 15x + -1x2 = 0 + x2 + -1x2 -23 + 15x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -23 + 15x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 23 + -15x + x2 = 0 Move the constant term to the right: Add '-23' to each side of the equation. 23 + -15x + -23 + x2 = 0 + -23 Reorder the terms: 23 + -23 + -15x + x2 = 0 + -23 Combine like terms: 23 + -23 = 0 0 + -15x + x2 = 0 + -23 -15x + x2 = 0 + -23 Combine like terms: 0 + -23 = -23 -15x + x2 = -23 The x term is -15x. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15x + 56.25 + x2 = -23 + 56.25 Reorder the terms: 56.25 + -15x + x2 = -23 + 56.25 Combine like terms: -23 + 56.25 = 33.25 56.25 + -15x + x2 = 33.25 Factor a perfect square on the left side: (x + -7.5)(x + -7.5) = 33.25 Calculate the square root of the right side: 5.766281297 Break this problem into two subproblems by setting (x + -7.5) equal to 5.766281297 and -5.766281297.Subproblem 1
x + -7.5 = 5.766281297 Simplifying x + -7.5 = 5.766281297 Reorder the terms: -7.5 + x = 5.766281297 Solving -7.5 + x = 5.766281297 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x = 5.766281297 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = 5.766281297 + 7.5 x = 5.766281297 + 7.5 Combine like terms: 5.766281297 + 7.5 = 13.266281297 x = 13.266281297 Simplifying x = 13.266281297Subproblem 2
x + -7.5 = -5.766281297 Simplifying x + -7.5 = -5.766281297 Reorder the terms: -7.5 + x = -5.766281297 Solving -7.5 + x = -5.766281297 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x = -5.766281297 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = -5.766281297 + 7.5 x = -5.766281297 + 7.5 Combine like terms: -5.766281297 + 7.5 = 1.733718703 x = 1.733718703 Simplifying x = 1.733718703Solution
The solution to the problem is based on the solutions from the subproblems. x = {13.266281297, 1.733718703}
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