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Simplifying (3x + 2)(4x + -2) = 16 Reorder the terms: (2 + 3x)(4x + -2) = 16 Reorder the terms: (2 + 3x)(-2 + 4x) = 16 Multiply (2 + 3x) * (-2 + 4x) (2(-2 + 4x) + 3x * (-2 + 4x)) = 16 ((-2 * 2 + 4x * 2) + 3x * (-2 + 4x)) = 16 ((-4 + 8x) + 3x * (-2 + 4x)) = 16 (-4 + 8x + (-2 * 3x + 4x * 3x)) = 16 (-4 + 8x + (-6x + 12x2)) = 16 Combine like terms: 8x + -6x = 2x (-4 + 2x + 12x2) = 16 Solving -4 + 2x + 12x2 = 16 Solving for variable 'x'. Reorder the terms: -4 + -16 + 2x + 12x2 = 16 + -16 Combine like terms: -4 + -16 = -20 -20 + 2x + 12x2 = 16 + -16 Combine like terms: 16 + -16 = 0 -20 + 2x + 12x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-10 + x + 6x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-10 + x + 6x2)' equal to zero and attempt to solve: Simplifying -10 + x + 6x2 = 0 Solving -10 + x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.666666667 + 0.1666666667x + x2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 0.1666666667x + 1.666666667 + x2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 0.1666666667x + x2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 0.1666666667x + x2 = 0 + 1.666666667 0.1666666667x + x2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 0.1666666667x + x2 = 1.666666667 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. 0.1666666667x + 0.25 + x2 = 1.666666667 + 0.25 Reorder the terms: 0.25 + 0.1666666667x + x2 = 1.666666667 + 0.25 Combine like terms: 1.666666667 + 0.25 = 1.916666667 0.25 + 0.1666666667x + x2 = 1.916666667 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 1.916666667 Calculate the square root of the right side: 1.384437311 Break this problem into two subproblems by setting (x + 0.5) equal to 1.384437311 and -1.384437311.Subproblem 1
x + 0.5 = 1.384437311 Simplifying x + 0.5 = 1.384437311 Reorder the terms: 0.5 + x = 1.384437311 Solving 0.5 + x = 1.384437311 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 = 1.384437311 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.384437311 + -0.5 x = 1.384437311 + -0.5 Combine like terms: 1.384437311 + -0.5 = 0.884437311 x = 0.884437311 Simplifying x = 0.884437311Subproblem 2
x + 0.5 = -1.384437311 Simplifying x + 0.5 = -1.384437311 Reorder the terms: 0.5 + x = -1.384437311 Solving 0.5 + x = -1.384437311 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 = -1.384437311 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.384437311 + -0.5 x = -1.384437311 + -0.5 Combine like terms: -1.384437311 + -0.5 = -1.884437311 x = -1.884437311 Simplifying x = -1.884437311Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.884437311, -1.884437311}Solution
x = {0.884437311, -1.884437311}
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