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Simplifying 2x4 + -78x3 + -135x2 = 0 Reorder the terms: -135x2 + -78x3 + 2x4 = 0 Solving -135x2 + -78x3 + 2x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(-135 + -78x + 2x2) = 0Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve: Simplifying x2 = 0 Solving x2 = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x2 = 0 Take the square root of each side: x = {0}Subproblem 2
Set the factor '(-135 + -78x + 2x2)' equal to zero and attempt to solve: Simplifying -135 + -78x + 2x2 = 0 Solving -135 + -78x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -67.5 + -39x + x2 = 0 Move the constant term to the right: Add '67.5' to each side of the equation. -67.5 + -39x + 67.5 + x2 = 0 + 67.5 Reorder the terms: -67.5 + 67.5 + -39x + x2 = 0 + 67.5 Combine like terms: -67.5 + 67.5 = 0.0 0.0 + -39x + x2 = 0 + 67.5 -39x + x2 = 0 + 67.5 Combine like terms: 0 + 67.5 = 67.5 -39x + x2 = 67.5 The x term is -39x. Take half its coefficient (-19.5). Square it (380.25) and add it to both sides. Add '380.25' to each side of the equation. -39x + 380.25 + x2 = 67.5 + 380.25 Reorder the terms: 380.25 + -39x + x2 = 67.5 + 380.25 Combine like terms: 67.5 + 380.25 = 447.75 380.25 + -39x + x2 = 447.75 Factor a perfect square on the left side: (x + -19.5)(x + -19.5) = 447.75 Calculate the square root of the right side: 21.160103969 Break this problem into two subproblems by setting (x + -19.5) equal to 21.160103969 and -21.160103969.Subproblem 1
x + -19.5 = 21.160103969 Simplifying x + -19.5 = 21.160103969 Reorder the terms: -19.5 + x = 21.160103969 Solving -19.5 + x = 21.160103969 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '19.5' to each side of the equation. -19.5 + 19.5 + x = 21.160103969 + 19.5 Combine like terms: -19.5 + 19.5 = 0.0 0.0 + x = 21.160103969 + 19.5 x = 21.160103969 + 19.5 Combine like terms: 21.160103969 + 19.5 = 40.660103969 x = 40.660103969 Simplifying x = 40.660103969Subproblem 2
x + -19.5 = -21.160103969 Simplifying x + -19.5 = -21.160103969 Reorder the terms: -19.5 + x = -21.160103969 Solving -19.5 + x = -21.160103969 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '19.5' to each side of the equation. -19.5 + 19.5 + x = -21.160103969 + 19.5 Combine like terms: -19.5 + 19.5 = 0.0 0.0 + x = -21.160103969 + 19.5 x = -21.160103969 + 19.5 Combine like terms: -21.160103969 + 19.5 = -1.660103969 x = -1.660103969 Simplifying x = -1.660103969Solution
The solution to the problem is based on the solutions from the subproblems. x = {40.660103969, -1.660103969}Solution
x = {0, 40.660103969, -1.660103969}
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