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Simplifying (4x + 3)(3x + 7) = x Reorder the terms: (3 + 4x)(3x + 7) = x Reorder the terms: (3 + 4x)(7 + 3x) = x Multiply (3 + 4x) * (7 + 3x) (3(7 + 3x) + 4x * (7 + 3x)) = x ((7 * 3 + 3x * 3) + 4x * (7 + 3x)) = x ((21 + 9x) + 4x * (7 + 3x)) = x (21 + 9x + (7 * 4x + 3x * 4x)) = x (21 + 9x + (28x + 12x2)) = x Combine like terms: 9x + 28x = 37x (21 + 37x + 12x2) = x Solving 21 + 37x + 12x2 = x Solving for variable 'x'. Reorder the terms: 21 + 37x + -1x + 12x2 = x + -1x Combine like terms: 37x + -1x = 36x 21 + 36x + 12x2 = x + -1x Combine like terms: x + -1x = 0 21 + 36x + 12x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(7 + 12x + 4x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(7 + 12x + 4x2)' equal to zero and attempt to solve: Simplifying 7 + 12x + 4x2 = 0 Solving 7 + 12x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.75 + 3x + x2 = 0 Move the constant term to the right: Add '-1.75' to each side of the equation. 1.75 + 3x + -1.75 + x2 = 0 + -1.75 Reorder the terms: 1.75 + -1.75 + 3x + x2 = 0 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + 3x + x2 = 0 + -1.75 3x + x2 = 0 + -1.75 Combine like terms: 0 + -1.75 = -1.75 3x + x2 = -1.75 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 = -1.75 + 2.25 Reorder the terms: 2.25 + 3x + x2 = -1.75 + 2.25 Combine like terms: -1.75 + 2.25 = 0.5 2.25 + 3x + x2 = 0.5 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (x + 1.5) equal to 0.707106781 and -0.707106781.Subproblem 1
x + 1.5 = 0.707106781 Simplifying x + 1.5 = 0.707106781 Reorder the terms: 1.5 + x = 0.707106781 Solving 1.5 + x = 0.707106781 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 = 0.707106781 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 0.707106781 + -1.5 x = 0.707106781 + -1.5 Combine like terms: 0.707106781 + -1.5 = -0.792893219 x = -0.792893219 Simplifying x = -0.792893219Subproblem 2
x + 1.5 = -0.707106781 Simplifying x + 1.5 = -0.707106781 Reorder the terms: 1.5 + x = -0.707106781 Solving 1.5 + x = -0.707106781 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 = -0.707106781 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -0.707106781 + -1.5 x = -0.707106781 + -1.5 Combine like terms: -0.707106781 + -1.5 = -2.207106781 x = -2.207106781 Simplifying x = -2.207106781Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.792893219, -2.207106781}Solution
x = {-0.792893219, -2.207106781}
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