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Simplifying (5x2 + 3x + -14) + -1(x2 + 15x + 9) = 0 Reorder the terms: (-14 + 3x + 5x2) + -1(x2 + 15x + 9) = 0 Remove parenthesis around (-14 + 3x + 5x2) -14 + 3x + 5x2 + -1(x2 + 15x + 9) = 0 Reorder the terms: -14 + 3x + 5x2 + -1(9 + 15x + x2) = 0 -14 + 3x + 5x2 + (9 * -1 + 15x * -1 + x2 * -1) = 0 -14 + 3x + 5x2 + (-9 + -15x + -1x2) = 0 Reorder the terms: -14 + -9 + 3x + -15x + 5x2 + -1x2 = 0 Combine like terms: -14 + -9 = -23 -23 + 3x + -15x + 5x2 + -1x2 = 0 Combine like terms: 3x + -15x = -12x -23 + -12x + 5x2 + -1x2 = 0 Combine like terms: 5x2 + -1x2 = 4x2 -23 + -12x + 4x2 = 0 Solving -23 + -12x + 4x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5.75 + -3x + x2 = 0 Move the constant term to the right: Add '5.75' to each side of the equation. -5.75 + -3x + 5.75 + x2 = 0 + 5.75 Reorder the terms: -5.75 + 5.75 + -3x + x2 = 0 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + -3x + x2 = 0 + 5.75 -3x + x2 = 0 + 5.75 Combine like terms: 0 + 5.75 = 5.75 -3x + x2 = 5.75 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = 5.75 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 5.75 + 2.25 Combine like terms: 5.75 + 2.25 = 8 2.25 + -3x + x2 = 8 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (x + -1.5) equal to 2.828427125 and -2.828427125.Subproblem 1
x + -1.5 = 2.828427125 Simplifying x + -1.5 = 2.828427125 Reorder the terms: -1.5 + x = 2.828427125 Solving -1.5 + x = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 2.828427125 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 2.828427125 + 1.5 x = 2.828427125 + 1.5 Combine like terms: 2.828427125 + 1.5 = 4.328427125 x = 4.328427125 Simplifying x = 4.328427125Subproblem 2
x + -1.5 = -2.828427125 Simplifying x + -1.5 = -2.828427125 Reorder the terms: -1.5 + x = -2.828427125 Solving -1.5 + x = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -2.828427125 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -2.828427125 + 1.5 x = -2.828427125 + 1.5 Combine like terms: -2.828427125 + 1.5 = -1.328427125 x = -1.328427125 Simplifying x = -1.328427125Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.328427125, -1.328427125}
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