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Simplifying 2x2 + 6x + -78 = 0 Reorder the terms: -78 + 6x + 2x2 = 0 Solving -78 + 6x + 2x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-39 + 3x + x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-39 + 3x + x2)' equal to zero and attempt to solve: Simplifying -39 + 3x + x2 = 0 Solving -39 + 3x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '39' to each side of the equation. -39 + 3x + 39 + x2 = 0 + 39 Reorder the terms: -39 + 39 + 3x + x2 = 0 + 39 Combine like terms: -39 + 39 = 0 0 + 3x + x2 = 0 + 39 3x + x2 = 0 + 39 Combine like terms: 0 + 39 = 39 3x + x2 = 39 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 = 39 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 39 + 2.25 Combine like terms: 39 + 2.25 = 41.25 2.25 + 3x + x2 = 41.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 41.25 Calculate the square root of the right side: 6.422616289 Break this problem into two subproblems by setting (x + 1.5) equal to 6.422616289 and -6.422616289.Subproblem 1
x + 1.5 = 6.422616289 Simplifying x + 1.5 = 6.422616289 Reorder the terms: 1.5 + x = 6.422616289 Solving 1.5 + x = 6.422616289 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 = 6.422616289 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 6.422616289 + -1.5 x = 6.422616289 + -1.5 Combine like terms: 6.422616289 + -1.5 = 4.922616289 x = 4.922616289 Simplifying x = 4.922616289Subproblem 2
x + 1.5 = -6.422616289 Simplifying x + 1.5 = -6.422616289 Reorder the terms: 1.5 + x = -6.422616289 Solving 1.5 + x = -6.422616289 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 = -6.422616289 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -6.422616289 + -1.5 x = -6.422616289 + -1.5 Combine like terms: -6.422616289 + -1.5 = -7.922616289 x = -7.922616289 Simplifying x = -7.922616289Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.922616289, -7.922616289}Solution
x = {4.922616289, -7.922616289}
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