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Simplifying 345x = 678 + -678x * x Multiply x * x 345x = 678 + -678x2 Solving 345x = 678 + -678x2 Solving for variable 'x'. Reorder the terms: -678 + 345x + 678x2 = 678 + -678x2 + -678 + 678x2 Reorder the terms: -678 + 345x + 678x2 = 678 + -678 + -678x2 + 678x2 Combine like terms: 678 + -678 = 0 -678 + 345x + 678x2 = 0 + -678x2 + 678x2 -678 + 345x + 678x2 = -678x2 + 678x2 Combine like terms: -678x2 + 678x2 = 0 -678 + 345x + 678x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-226 + 115x + 226x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-226 + 115x + 226x2)' equal to zero and attempt to solve: Simplifying -226 + 115x + 226x2 = 0 Solving -226 + 115x + 226x2 = 0 Begin completing the square. Divide all terms by 226 the coefficient of the squared term: Divide each side by '226'. -1 + 0.5088495575x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.5088495575x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + 0.5088495575x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.5088495575x + x2 = 0 + 1 0.5088495575x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.5088495575x + x2 = 1 The x term is 0.5088495575x. Take half its coefficient (0.2544247788). Square it (0.06473196807) and add it to both sides. Add '0.06473196807' to each side of the equation. 0.5088495575x + 0.06473196807 + x2 = 1 + 0.06473196807 Reorder the terms: 0.06473196807 + 0.5088495575x + x2 = 1 + 0.06473196807 Combine like terms: 1 + 0.06473196807 = 1.06473196807 0.06473196807 + 0.5088495575x + x2 = 1.06473196807 Factor a perfect square on the left side: (x + 0.2544247788)(x + 0.2544247788) = 1.06473196807 Calculate the square root of the right side: 1.031858502 Break this problem into two subproblems by setting (x + 0.2544247788) equal to 1.031858502 and -1.031858502.Subproblem 1
x + 0.2544247788 = 1.031858502 Simplifying x + 0.2544247788 = 1.031858502 Reorder the terms: 0.2544247788 + x = 1.031858502 Solving 0.2544247788 + x = 1.031858502 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2544247788' to each side of the equation. 0.2544247788 + -0.2544247788 + x = 1.031858502 + -0.2544247788 Combine like terms: 0.2544247788 + -0.2544247788 = 0.0000000000 0.0000000000 + x = 1.031858502 + -0.2544247788 x = 1.031858502 + -0.2544247788 Combine like terms: 1.031858502 + -0.2544247788 = 0.7774337232 x = 0.7774337232 Simplifying x = 0.7774337232Subproblem 2
x + 0.2544247788 = -1.031858502 Simplifying x + 0.2544247788 = -1.031858502 Reorder the terms: 0.2544247788 + x = -1.031858502 Solving 0.2544247788 + x = -1.031858502 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2544247788' to each side of the equation. 0.2544247788 + -0.2544247788 + x = -1.031858502 + -0.2544247788 Combine like terms: 0.2544247788 + -0.2544247788 = 0.0000000000 0.0000000000 + x = -1.031858502 + -0.2544247788 x = -1.031858502 + -0.2544247788 Combine like terms: -1.031858502 + -0.2544247788 = -1.2862832808 x = -1.2862832808 Simplifying x = -1.2862832808Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7774337232, -1.2862832808}Solution
x = {0.7774337232, -1.2862832808}
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