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Simplifying 4x2 + 16x + 16 = 4x + 24 Reorder the terms: 16 + 16x + 4x2 = 4x + 24 Reorder the terms: 16 + 16x + 4x2 = 24 + 4x Solving 16 + 16x + 4x2 = 24 + 4x Solving for variable 'x'. Reorder the terms: 16 + -24 + 16x + -4x + 4x2 = 24 + 4x + -24 + -4x Combine like terms: 16 + -24 = -8 -8 + 16x + -4x + 4x2 = 24 + 4x + -24 + -4x Combine like terms: 16x + -4x = 12x -8 + 12x + 4x2 = 24 + 4x + -24 + -4x Reorder the terms: -8 + 12x + 4x2 = 24 + -24 + 4x + -4x Combine like terms: 24 + -24 = 0 -8 + 12x + 4x2 = 0 + 4x + -4x -8 + 12x + 4x2 = 4x + -4x Combine like terms: 4x + -4x = 0 -8 + 12x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-2 + 3x + x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-2 + 3x + x2)' equal to zero and attempt to solve: Simplifying -2 + 3x + x2 = 0 Solving -2 + 3x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + 3x + 2 + x2 = 0 + 2 Reorder the terms: -2 + 2 + 3x + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 3x + x2 = 0 + 2 3x + x2 = 0 + 2 Combine like terms: 0 + 2 = 2 3x + x2 = 2 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 = 2 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 2 + 2.25 Combine like terms: 2 + 2.25 = 4.25 2.25 + 3x + x2 = 4.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (x + 1.5) equal to 2.061552813 and -2.061552813.Subproblem 1
x + 1.5 = 2.061552813 Simplifying x + 1.5 = 2.061552813 Reorder the terms: 1.5 + x = 2.061552813 Solving 1.5 + x = 2.061552813 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 = 2.061552813 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 2.061552813 + -1.5 x = 2.061552813 + -1.5 Combine like terms: 2.061552813 + -1.5 = 0.561552813 x = 0.561552813 Simplifying x = 0.561552813Subproblem 2
x + 1.5 = -2.061552813 Simplifying x + 1.5 = -2.061552813 Reorder the terms: 1.5 + x = -2.061552813 Solving 1.5 + x = -2.061552813 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 = -2.061552813 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -2.061552813 + -1.5 x = -2.061552813 + -1.5 Combine like terms: -2.061552813 + -1.5 = -3.561552813 x = -3.561552813 Simplifying x = -3.561552813Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.561552813, -3.561552813}Solution
x = {0.561552813, -3.561552813}
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