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Simplifying 0 = 4(x2 + 3x + 1) Reorder the terms: 0 = 4(1 + 3x + x2) 0 = (1 * 4 + 3x * 4 + x2 * 4) 0 = (4 + 12x + 4x2) Solving 0 = 4 + 12x + 4x2 Solving for variable 'x'. Combine like terms: 0 + -4 = -4 -4 + -12x + -4x2 = 4 + 12x + 4x2 + -4 + -12x + -4x2 Reorder the terms: -4 + -12x + -4x2 = 4 + -4 + 12x + -12x + 4x2 + -4x2 Combine like terms: 4 + -4 = 0 -4 + -12x + -4x2 = 0 + 12x + -12x + 4x2 + -4x2 -4 + -12x + -4x2 = 12x + -12x + 4x2 + -4x2 Combine like terms: 12x + -12x = 0 -4 + -12x + -4x2 = 0 + 4x2 + -4x2 -4 + -12x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 -4 + -12x + -4x2 = 0 Factor out the Greatest Common Factor (GCF), '-4'. -4(1 + 3x + x2) = 0 Ignore the factor -4.Subproblem 1
Set the factor '(1 + 3x + x2)' equal to zero and attempt to solve: Simplifying 1 + 3x + x2 = 0 Solving 1 + 3x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 3x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + 3x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 3x + x2 = 0 + -1 3x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 3x + x2 = -1 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 + 2.25 Reorder the terms: 2.25 + 3x + x2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + 3x + x2 = 1.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (x + 1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
x + 1.5 = 1.118033989 Simplifying x + 1.5 = 1.118033989 Reorder the terms: 1.5 + x = 1.118033989 Solving 1.5 + x = 1.118033989 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 = 1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.118033989 + -1.5 x = 1.118033989 + -1.5 Combine like terms: 1.118033989 + -1.5 = -0.381966011 x = -0.381966011 Simplifying x = -0.381966011Subproblem 2
x + 1.5 = -1.118033989 Simplifying x + 1.5 = -1.118033989 Reorder the terms: 1.5 + x = -1.118033989 Solving 1.5 + x = -1.118033989 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 = -1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.118033989 + -1.5 x = -1.118033989 + -1.5 Combine like terms: -1.118033989 + -1.5 = -2.618033989 x = -2.618033989 Simplifying x = -2.618033989Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.381966011, -2.618033989}Solution
x = {-0.381966011, -2.618033989}
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