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Simplifying -18x2 + 6x + 11 = 0 Reorder the terms: 11 + 6x + -18x2 = 0 Solving 11 + 6x + -18x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -18 the coefficient of the squared term: Divide each side by '-18'. -0.6111111111 + -0.3333333333x + x2 = 0 Move the constant term to the right: Add '0.6111111111' to each side of the equation. -0.6111111111 + -0.3333333333x + 0.6111111111 + x2 = 0 + 0.6111111111 Reorder the terms: -0.6111111111 + 0.6111111111 + -0.3333333333x + x2 = 0 + 0.6111111111 Combine like terms: -0.6111111111 + 0.6111111111 = 0.0000000000 0.0000000000 + -0.3333333333x + x2 = 0 + 0.6111111111 -0.3333333333x + x2 = 0 + 0.6111111111 Combine like terms: 0 + 0.6111111111 = 0.6111111111 -0.3333333333x + x2 = 0.6111111111 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 = 0.6111111111 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333x + x2 = 0.6111111111 + 0.02777777779 Combine like terms: 0.6111111111 + 0.02777777779 = 0.63888888889 0.02777777779 + -0.3333333333x + x2 = 0.63888888889 Factor a perfect square on the left side: (x + -0.1666666667)(x + -0.1666666667) = 0.63888888889 Calculate the square root of the right side: 0.799305254 Break this problem into two subproblems by setting (x + -0.1666666667) equal to 0.799305254 and -0.799305254.Subproblem 1
x + -0.1666666667 = 0.799305254 Simplifying x + -0.1666666667 = 0.799305254 Reorder the terms: -0.1666666667 + x = 0.799305254 Solving -0.1666666667 + x = 0.799305254 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 = 0.799305254 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + x = 0.799305254 + 0.1666666667 x = 0.799305254 + 0.1666666667 Combine like terms: 0.799305254 + 0.1666666667 = 0.9659719207 x = 0.9659719207 Simplifying x = 0.9659719207Subproblem 2
x + -0.1666666667 = -0.799305254 Simplifying x + -0.1666666667 = -0.799305254 Reorder the terms: -0.1666666667 + x = -0.799305254 Solving -0.1666666667 + x = -0.799305254 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 = -0.799305254 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + x = -0.799305254 + 0.1666666667 x = -0.799305254 + 0.1666666667 Combine like terms: -0.799305254 + 0.1666666667 = -0.6326385873 x = -0.6326385873 Simplifying x = -0.6326385873Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.9659719207, -0.6326385873}
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