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Simplifying (2x + -5)(3x + -2) + 7x = 22 Reorder the terms: (-5 + 2x)(3x + -2) + 7x = 22 Reorder the terms: (-5 + 2x)(-2 + 3x) + 7x = 22 Multiply (-5 + 2x) * (-2 + 3x) (-5(-2 + 3x) + 2x * (-2 + 3x)) + 7x = 22 ((-2 * -5 + 3x * -5) + 2x * (-2 + 3x)) + 7x = 22 ((10 + -15x) + 2x * (-2 + 3x)) + 7x = 22 (10 + -15x + (-2 * 2x + 3x * 2x)) + 7x = 22 (10 + -15x + (-4x + 6x2)) + 7x = 22 Combine like terms: -15x + -4x = -19x (10 + -19x + 6x2) + 7x = 22 Reorder the terms: 10 + -19x + 7x + 6x2 = 22 Combine like terms: -19x + 7x = -12x 10 + -12x + 6x2 = 22 Solving 10 + -12x + 6x2 = 22 Solving for variable 'x'. Reorder the terms: 10 + -22 + -12x + 6x2 = 22 + -22 Combine like terms: 10 + -22 = -12 -12 + -12x + 6x2 = 22 + -22 Combine like terms: 22 + -22 = 0 -12 + -12x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-2 + -2x + x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-2 + -2x + x2)' equal to zero and attempt to solve: Simplifying -2 + -2x + x2 = 0 Solving -2 + -2x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + -2x + 2 + x2 = 0 + 2 Reorder the terms: -2 + 2 + -2x + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -2x + x2 = 0 + 2 -2x + x2 = 0 + 2 Combine like terms: 0 + 2 = 2 -2x + x2 = 2 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = 2 + 1 Reorder the terms: 1 + -2x + x2 = 2 + 1 Combine like terms: 2 + 1 = 3 1 + -2x + x2 = 3 Factor a perfect square on the left side: (x + -1)(x + -1) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (x + -1) equal to 1.732050808 and -1.732050808.Subproblem 1
x + -1 = 1.732050808 Simplifying x + -1 = 1.732050808 Reorder the terms: -1 + x = 1.732050808 Solving -1 + x = 1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 1.732050808 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.732050808 + 1 x = 1.732050808 + 1 Combine like terms: 1.732050808 + 1 = 2.732050808 x = 2.732050808 Simplifying x = 2.732050808Subproblem 2
x + -1 = -1.732050808 Simplifying x + -1 = -1.732050808 Reorder the terms: -1 + x = -1.732050808 Solving -1 + x = -1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -1.732050808 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.732050808 + 1 x = -1.732050808 + 1 Combine like terms: -1.732050808 + 1 = -0.732050808 x = -0.732050808 Simplifying x = -0.732050808Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.732050808, -0.732050808}Solution
x = {2.732050808, -0.732050808}
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