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Simplifying 0 = 20x2 + -54x + 18 Reorder the terms: 0 = 18 + -54x + 20x2 Solving 0 = 18 + -54x + 20x2 Solving for variable 'x'. Combine like terms: 0 + -18 = -18 -18 + 54x + -20x2 = 18 + -54x + 20x2 + -18 + 54x + -20x2 Reorder the terms: -18 + 54x + -20x2 = 18 + -18 + -54x + 54x + 20x2 + -20x2 Combine like terms: 18 + -18 = 0 -18 + 54x + -20x2 = 0 + -54x + 54x + 20x2 + -20x2 -18 + 54x + -20x2 = -54x + 54x + 20x2 + -20x2 Combine like terms: -54x + 54x = 0 -18 + 54x + -20x2 = 0 + 20x2 + -20x2 -18 + 54x + -20x2 = 20x2 + -20x2 Combine like terms: 20x2 + -20x2 = 0 -18 + 54x + -20x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-9 + 27x + -10x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-9 + 27x + -10x2)' equal to zero and attempt to solve: Simplifying -9 + 27x + -10x2 = 0 Solving -9 + 27x + -10x2 = 0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. 0.9 + -2.7x + x2 = 0.0 Move the constant term to the right: Add '-0.9' to each side of the equation. 0.9 + -2.7x + -0.9 + x2 = 0.0 + -0.9 Reorder the terms: 0.9 + -0.9 + -2.7x + x2 = 0.0 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + -2.7x + x2 = 0.0 + -0.9 -2.7x + x2 = 0.0 + -0.9 Combine like terms: 0.0 + -0.9 = -0.9 -2.7x + x2 = -0.9 The x term is -2.7x. Take half its coefficient (-1.35). Square it (1.8225) and add it to both sides. Add '1.8225' to each side of the equation. -2.7x + 1.8225 + x2 = -0.9 + 1.8225 Reorder the terms: 1.8225 + -2.7x + x2 = -0.9 + 1.8225 Combine like terms: -0.9 + 1.8225 = 0.9225 1.8225 + -2.7x + x2 = 0.9225 Factor a perfect square on the left side: (x + -1.35)(x + -1.35) = 0.9225 Calculate the square root of the right side: 0.960468636 Break this problem into two subproblems by setting (x + -1.35) equal to 0.960468636 and -0.960468636.Subproblem 1
x + -1.35 = 0.960468636 Simplifying x + -1.35 = 0.960468636 Reorder the terms: -1.35 + x = 0.960468636 Solving -1.35 + x = 0.960468636 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.35' to each side of the equation. -1.35 + 1.35 + x = 0.960468636 + 1.35 Combine like terms: -1.35 + 1.35 = 0.00 0.00 + x = 0.960468636 + 1.35 x = 0.960468636 + 1.35 Combine like terms: 0.960468636 + 1.35 = 2.310468636 x = 2.310468636 Simplifying x = 2.310468636Subproblem 2
x + -1.35 = -0.960468636 Simplifying x + -1.35 = -0.960468636 Reorder the terms: -1.35 + x = -0.960468636 Solving -1.35 + x = -0.960468636 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.35' to each side of the equation. -1.35 + 1.35 + x = -0.960468636 + 1.35 Combine like terms: -1.35 + 1.35 = 0.00 0.00 + x = -0.960468636 + 1.35 x = -0.960468636 + 1.35 Combine like terms: -0.960468636 + 1.35 = 0.389531364 x = 0.389531364 Simplifying x = 0.389531364Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.310468636, 0.389531364}Solution
x = {2.310468636, 0.389531364}
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