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Simplifying 4x + -1(2x + 3) + -1(3x + -5) = 49 + -1(6x + -1)(1 + 2x) Reorder the terms: 4x + -1(3 + 2x) + -1(3x + -5) = 49 + -1(6x + -1)(1 + 2x) 4x + (3 * -1 + 2x * -1) + -1(3x + -5) = 49 + -1(6x + -1)(1 + 2x) 4x + (-3 + -2x) + -1(3x + -5) = 49 + -1(6x + -1)(1 + 2x) Reorder the terms: 4x + -3 + -2x + -1(-5 + 3x) = 49 + -1(6x + -1)(1 + 2x) 4x + -3 + -2x + (-5 * -1 + 3x * -1) = 49 + -1(6x + -1)(1 + 2x) 4x + -3 + -2x + (5 + -3x) = 49 + -1(6x + -1)(1 + 2x) Reorder the terms: -3 + 5 + 4x + -2x + -3x = 49 + -1(6x + -1)(1 + 2x) Combine like terms: -3 + 5 = 2 2 + 4x + -2x + -3x = 49 + -1(6x + -1)(1 + 2x) Combine like terms: 4x + -2x = 2x 2 + 2x + -3x = 49 + -1(6x + -1)(1 + 2x) Combine like terms: 2x + -3x = -1x 2 + -1x = 49 + -1(6x + -1)(1 + 2x) Reorder the terms: 2 + -1x = 49 + -1(-1 + 6x)(1 + 2x) Multiply (-1 + 6x) * (1 + 2x) 2 + -1x = 49 + -1(-1(1 + 2x) + 6x * (1 + 2x)) 2 + -1x = 49 + -1((1 * -1 + 2x * -1) + 6x * (1 + 2x)) 2 + -1x = 49 + -1((-1 + -2x) + 6x * (1 + 2x)) 2 + -1x = 49 + -1(-1 + -2x + (1 * 6x + 2x * 6x)) 2 + -1x = 49 + -1(-1 + -2x + (6x + 12x2)) Combine like terms: -2x + 6x = 4x 2 + -1x = 49 + -1(-1 + 4x + 12x2) 2 + -1x = 49 + (-1 * -1 + 4x * -1 + 12x2 * -1) 2 + -1x = 49 + (1 + -4x + -12x2) Combine like terms: 49 + 1 = 50 2 + -1x = 50 + -4x + -12x2 Solving 2 + -1x = 50 + -4x + -12x2 Solving for variable 'x'. Reorder the terms: 2 + -50 + -1x + 4x + 12x2 = 50 + -4x + -12x2 + -50 + 4x + 12x2 Combine like terms: 2 + -50 = -48 -48 + -1x + 4x + 12x2 = 50 + -4x + -12x2 + -50 + 4x + 12x2 Combine like terms: -1x + 4x = 3x -48 + 3x + 12x2 = 50 + -4x + -12x2 + -50 + 4x + 12x2 Reorder the terms: -48 + 3x + 12x2 = 50 + -50 + -4x + 4x + -12x2 + 12x2 Combine like terms: 50 + -50 = 0 -48 + 3x + 12x2 = 0 + -4x + 4x + -12x2 + 12x2 -48 + 3x + 12x2 = -4x + 4x + -12x2 + 12x2 Combine like terms: -4x + 4x = 0 -48 + 3x + 12x2 = 0 + -12x2 + 12x2 -48 + 3x + 12x2 = -12x2 + 12x2 Combine like terms: -12x2 + 12x2 = 0 -48 + 3x + 12x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-16 + x + 4x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-16 + x + 4x2)' equal to zero and attempt to solve: Simplifying -16 + x + 4x2 = 0 Solving -16 + x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -4 + 0.25x + x2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 0.25x + 4 + x2 = 0 + 4 Reorder the terms: -4 + 4 + 0.25x + x2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 0.25x + x2 = 0 + 4 0.25x + x2 = 0 + 4 Combine like terms: 0 + 4 = 4 0.25x + x2 = 4 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.25x + 0.25 + x2 = 4 + 0.25 Reorder the terms: 0.25 + 0.25x + x2 = 4 + 0.25 Combine like terms: 4 + 0.25 = 4.25 0.25 + 0.25x + x2 = 4.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (x + 0.5) equal to 2.061552813 and -2.061552813.Subproblem 1
x + 0.5 = 2.061552813 Simplifying x + 0.5 = 2.061552813 Reorder the terms: 0.5 + x = 2.061552813 Solving 0.5 + x = 2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.061552813 + -0.5 x = 2.061552813 + -0.5 Combine like terms: 2.061552813 + -0.5 = 1.561552813 x = 1.561552813 Simplifying x = 1.561552813Subproblem 2
x + 0.5 = -2.061552813 Simplifying x + 0.5 = -2.061552813 Reorder the terms: 0.5 + x = -2.061552813 Solving 0.5 + x = -2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.061552813 + -0.5 x = -2.061552813 + -0.5 Combine like terms: -2.061552813 + -0.5 = -2.561552813 x = -2.561552813 Simplifying x = -2.561552813Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.561552813, -2.561552813}Solution
x = {1.561552813, -2.561552813}
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