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Simplifying (5x + 4)(3x + 2) = (7x + -1)(3x + -5) + 60 + -51 Reorder the terms: (4 + 5x)(3x + 2) = (7x + -1)(3x + -5) + 60 + -51 Reorder the terms: (4 + 5x)(2 + 3x) = (7x + -1)(3x + -5) + 60 + -51 Multiply (4 + 5x) * (2 + 3x) (4(2 + 3x) + 5x * (2 + 3x)) = (7x + -1)(3x + -5) + 60 + -51 ((2 * 4 + 3x * 4) + 5x * (2 + 3x)) = (7x + -1)(3x + -5) + 60 + -51 ((8 + 12x) + 5x * (2 + 3x)) = (7x + -1)(3x + -5) + 60 + -51 (8 + 12x + (2 * 5x + 3x * 5x)) = (7x + -1)(3x + -5) + 60 + -51 (8 + 12x + (10x + 15x2)) = (7x + -1)(3x + -5) + 60 + -51 Combine like terms: 12x + 10x = 22x (8 + 22x + 15x2) = (7x + -1)(3x + -5) + 60 + -51 Reorder the terms: 8 + 22x + 15x2 = (-1 + 7x)(3x + -5) + 60 + -51 Reorder the terms: 8 + 22x + 15x2 = (-1 + 7x)(-5 + 3x) + 60 + -51 Multiply (-1 + 7x) * (-5 + 3x) 8 + 22x + 15x2 = (-1(-5 + 3x) + 7x * (-5 + 3x)) + 60 + -51 8 + 22x + 15x2 = ((-5 * -1 + 3x * -1) + 7x * (-5 + 3x)) + 60 + -51 8 + 22x + 15x2 = ((5 + -3x) + 7x * (-5 + 3x)) + 60 + -51 8 + 22x + 15x2 = (5 + -3x + (-5 * 7x + 3x * 7x)) + 60 + -51 8 + 22x + 15x2 = (5 + -3x + (-35x + 21x2)) + 60 + -51 Combine like terms: -3x + -35x = -38x 8 + 22x + 15x2 = (5 + -38x + 21x2) + 60 + -51 Reorder the terms: 8 + 22x + 15x2 = 5 + 60 + -51 + -38x + 21x2 Combine like terms: 5 + 60 = 65 8 + 22x + 15x2 = 65 + -51 + -38x + 21x2 Combine like terms: 65 + -51 = 14 8 + 22x + 15x2 = 14 + -38x + 21x2 Solving 8 + 22x + 15x2 = 14 + -38x + 21x2 Solving for variable 'x'. Reorder the terms: 8 + -14 + 22x + 38x + 15x2 + -21x2 = 14 + -38x + 21x2 + -14 + 38x + -21x2 Combine like terms: 8 + -14 = -6 -6 + 22x + 38x + 15x2 + -21x2 = 14 + -38x + 21x2 + -14 + 38x + -21x2 Combine like terms: 22x + 38x = 60x -6 + 60x + 15x2 + -21x2 = 14 + -38x + 21x2 + -14 + 38x + -21x2 Combine like terms: 15x2 + -21x2 = -6x2 -6 + 60x + -6x2 = 14 + -38x + 21x2 + -14 + 38x + -21x2 Reorder the terms: -6 + 60x + -6x2 = 14 + -14 + -38x + 38x + 21x2 + -21x2 Combine like terms: 14 + -14 = 0 -6 + 60x + -6x2 = 0 + -38x + 38x + 21x2 + -21x2 -6 + 60x + -6x2 = -38x + 38x + 21x2 + -21x2 Combine like terms: -38x + 38x = 0 -6 + 60x + -6x2 = 0 + 21x2 + -21x2 -6 + 60x + -6x2 = 21x2 + -21x2 Combine like terms: 21x2 + -21x2 = 0 -6 + 60x + -6x2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-1 + 10x + -1x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-1 + 10x + -1x2)' equal to zero and attempt to solve: Simplifying -1 + 10x + -1x2 = 0 Solving -1 + 10x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1 + -10x + x2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -10x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + -10x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -10x + x2 = 0 + -1 -10x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 -10x + x2 = -1 The x term is -10x. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10x + 25 + x2 = -1 + 25 Reorder the terms: 25 + -10x + x2 = -1 + 25 Combine like terms: -1 + 25 = 24 25 + -10x + x2 = 24 Factor a perfect square on the left side: (x + -5)(x + -5) = 24 Calculate the square root of the right side: 4.898979486 Break this problem into two subproblems by setting (x + -5) equal to 4.898979486 and -4.898979486.Subproblem 1
x + -5 = 4.898979486 Simplifying x + -5 = 4.898979486 Reorder the terms: -5 + x = 4.898979486 Solving -5 + x = 4.898979486 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 4.898979486 + 5 Combine like terms: -5 + 5 = 0 0 + x = 4.898979486 + 5 x = 4.898979486 + 5 Combine like terms: 4.898979486 + 5 = 9.898979486 x = 9.898979486 Simplifying x = 9.898979486Subproblem 2
x + -5 = -4.898979486 Simplifying x + -5 = -4.898979486 Reorder the terms: -5 + x = -4.898979486 Solving -5 + x = -4.898979486 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -4.898979486 + 5 Combine like terms: -5 + 5 = 0 0 + x = -4.898979486 + 5 x = -4.898979486 + 5 Combine like terms: -4.898979486 + 5 = 0.101020514 x = 0.101020514 Simplifying x = 0.101020514Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.898979486, 0.101020514}Solution
x = {9.898979486, 0.101020514}
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