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Simplifying -16x2 + 54x + 18 = 0 Reorder the terms: 18 + 54x + -16x2 = 0 Solving 18 + 54x + -16x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(9 + 27x + -8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(9 + 27x + -8x2)' equal to zero and attempt to solve: Simplifying 9 + 27x + -8x2 = 0 Solving 9 + 27x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1.125 + -3.375x + x2 = 0 Move the constant term to the right: Add '1.125' to each side of the equation. -1.125 + -3.375x + 1.125 + x2 = 0 + 1.125 Reorder the terms: -1.125 + 1.125 + -3.375x + x2 = 0 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + -3.375x + x2 = 0 + 1.125 -3.375x + x2 = 0 + 1.125 Combine like terms: 0 + 1.125 = 1.125 -3.375x + x2 = 1.125 The x term is -3.375x. Take half its coefficient (-1.6875). Square it (2.84765625) and add it to both sides. Add '2.84765625' to each side of the equation. -3.375x + 2.84765625 + x2 = 1.125 + 2.84765625 Reorder the terms: 2.84765625 + -3.375x + x2 = 1.125 + 2.84765625 Combine like terms: 1.125 + 2.84765625 = 3.97265625 2.84765625 + -3.375x + x2 = 3.97265625 Factor a perfect square on the left side: (x + -1.6875)(x + -1.6875) = 3.97265625 Calculate the square root of the right side: 1.99315234 Break this problem into two subproblems by setting (x + -1.6875) equal to 1.99315234 and -1.99315234.Subproblem 1
x + -1.6875 = 1.99315234 Simplifying x + -1.6875 = 1.99315234 Reorder the terms: -1.6875 + x = 1.99315234 Solving -1.6875 + x = 1.99315234 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.6875' to each side of the equation. -1.6875 + 1.6875 + x = 1.99315234 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + x = 1.99315234 + 1.6875 x = 1.99315234 + 1.6875 Combine like terms: 1.99315234 + 1.6875 = 3.68065234 x = 3.68065234 Simplifying x = 3.68065234Subproblem 2
x + -1.6875 = -1.99315234 Simplifying x + -1.6875 = -1.99315234 Reorder the terms: -1.6875 + x = -1.99315234 Solving -1.6875 + x = -1.99315234 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.6875' to each side of the equation. -1.6875 + 1.6875 + x = -1.99315234 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + x = -1.99315234 + 1.6875 x = -1.99315234 + 1.6875 Combine like terms: -1.99315234 + 1.6875 = -0.30565234 x = -0.30565234 Simplifying x = -0.30565234Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.68065234, -0.30565234}Solution
x = {3.68065234, -0.30565234}
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