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Simplifying 18x + 130x2 = 54 Solving 18x + 130x2 = 54 Solving for variable 'x'. Reorder the terms: -54 + 18x + 130x2 = 54 + -54 Combine like terms: 54 + -54 = 0 -54 + 18x + 130x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-27 + 9x + 65x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-27 + 9x + 65x2)' equal to zero and attempt to solve: Simplifying -27 + 9x + 65x2 = 0 Solving -27 + 9x + 65x2 = 0 Begin completing the square. Divide all terms by 65 the coefficient of the squared term: Divide each side by '65'. -0.4153846154 + 0.1384615385x + x2 = 0 Move the constant term to the right: Add '0.4153846154' to each side of the equation. -0.4153846154 + 0.1384615385x + 0.4153846154 + x2 = 0 + 0.4153846154 Reorder the terms: -0.4153846154 + 0.4153846154 + 0.1384615385x + x2 = 0 + 0.4153846154 Combine like terms: -0.4153846154 + 0.4153846154 = 0.0000000000 0.0000000000 + 0.1384615385x + x2 = 0 + 0.4153846154 0.1384615385x + x2 = 0 + 0.4153846154 Combine like terms: 0 + 0.4153846154 = 0.4153846154 0.1384615385x + x2 = 0.4153846154 The x term is 0.1384615385x. Take half its coefficient (0.06923076925). Square it (0.004792899411) and add it to both sides. Add '0.004792899411' to each side of the equation. 0.1384615385x + 0.004792899411 + x2 = 0.4153846154 + 0.004792899411 Reorder the terms: 0.004792899411 + 0.1384615385x + x2 = 0.4153846154 + 0.004792899411 Combine like terms: 0.4153846154 + 0.004792899411 = 0.420177514811 0.004792899411 + 0.1384615385x + x2 = 0.420177514811 Factor a perfect square on the left side: (x + 0.06923076925)(x + 0.06923076925) = 0.420177514811 Calculate the square root of the right side: 0.648211011 Break this problem into two subproblems by setting (x + 0.06923076925) equal to 0.648211011 and -0.648211011.Subproblem 1
x + 0.06923076925 = 0.648211011 Simplifying x + 0.06923076925 = 0.648211011 Reorder the terms: 0.06923076925 + x = 0.648211011 Solving 0.06923076925 + x = 0.648211011 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.06923076925' to each side of the equation. 0.06923076925 + -0.06923076925 + x = 0.648211011 + -0.06923076925 Combine like terms: 0.06923076925 + -0.06923076925 = 0.00000000000 0.00000000000 + x = 0.648211011 + -0.06923076925 x = 0.648211011 + -0.06923076925 Combine like terms: 0.648211011 + -0.06923076925 = 0.57898024175 x = 0.57898024175 Simplifying x = 0.57898024175Subproblem 2
x + 0.06923076925 = -0.648211011 Simplifying x + 0.06923076925 = -0.648211011 Reorder the terms: 0.06923076925 + x = -0.648211011 Solving 0.06923076925 + x = -0.648211011 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.06923076925' to each side of the equation. 0.06923076925 + -0.06923076925 + x = -0.648211011 + -0.06923076925 Combine like terms: 0.06923076925 + -0.06923076925 = 0.00000000000 0.00000000000 + x = -0.648211011 + -0.06923076925 x = -0.648211011 + -0.06923076925 Combine like terms: -0.648211011 + -0.06923076925 = -0.71744178025 x = -0.71744178025 Simplifying x = -0.71744178025Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.57898024175, -0.71744178025}Solution
x = {0.57898024175, -0.71744178025}
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