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Simplifying 54x2 + 18x + -4 = 0 Reorder the terms: -4 + 18x + 54x2 = 0 Solving -4 + 18x + 54x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-2 + 9x + 27x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-2 + 9x + 27x2)' equal to zero and attempt to solve: Simplifying -2 + 9x + 27x2 = 0 Solving -2 + 9x + 27x2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.07407407407 + 0.3333333333x + x2 = 0 Move the constant term to the right: Add '0.07407407407' to each side of the equation. -0.07407407407 + 0.3333333333x + 0.07407407407 + x2 = 0 + 0.07407407407 Reorder the terms: -0.07407407407 + 0.07407407407 + 0.3333333333x + x2 = 0 + 0.07407407407 Combine like terms: -0.07407407407 + 0.07407407407 = 0.00000000000 0.00000000000 + 0.3333333333x + x2 = 0 + 0.07407407407 0.3333333333x + x2 = 0 + 0.07407407407 Combine like terms: 0 + 0.07407407407 = 0.07407407407 0.3333333333x + x2 = 0.07407407407 The x term is 0.3333333333x. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333x + 0.02777777779 + x2 = 0.07407407407 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333x + x2 = 0.07407407407 + 0.02777777779 Combine like terms: 0.07407407407 + 0.02777777779 = 0.10185185186 0.02777777779 + 0.3333333333x + x2 = 0.10185185186 Factor a perfect square on the left side: (x + 0.1666666667)(x + 0.1666666667) = 0.10185185186 Calculate the square root of the right side: 0.319142369 Break this problem into two subproblems by setting (x + 0.1666666667) equal to 0.319142369 and -0.319142369.Subproblem 1
x + 0.1666666667 = 0.319142369 Simplifying x + 0.1666666667 = 0.319142369 Reorder the terms: 0.1666666667 + x = 0.319142369 Solving 0.1666666667 + x = 0.319142369 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = 0.319142369 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 0.319142369 + -0.1666666667 x = 0.319142369 + -0.1666666667 Combine like terms: 0.319142369 + -0.1666666667 = 0.1524757023 x = 0.1524757023 Simplifying x = 0.1524757023Subproblem 2
x + 0.1666666667 = -0.319142369 Simplifying x + 0.1666666667 = -0.319142369 Reorder the terms: 0.1666666667 + x = -0.319142369 Solving 0.1666666667 + x = -0.319142369 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = -0.319142369 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -0.319142369 + -0.1666666667 x = -0.319142369 + -0.1666666667 Combine like terms: -0.319142369 + -0.1666666667 = -0.4858090357 x = -0.4858090357 Simplifying x = -0.4858090357Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.1524757023, -0.4858090357}Solution
x = {0.1524757023, -0.4858090357}
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