18x^2-77x+18=0

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Solution for 18x^2-77x+18=0 equation:


Simplifying
18x2 + -77x + 18 = 0

Reorder the terms:
18 + -77x + 18x2 = 0

Solving
18 + -77x + 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'.
1 + -4.277777778x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -4.277777778x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -4.277777778x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -4.277777778x + x2 = 0 + -1
-4.277777778x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-4.277777778x + x2 = -1

The x term is -4.277777778x.  Take half its coefficient (-2.138888889).
Square it (4.574845679) and add it to both sides.

Add '4.574845679' to each side of the equation.
-4.277777778x + 4.574845679 + x2 = -1 + 4.574845679

Reorder the terms:
4.574845679 + -4.277777778x + x2 = -1 + 4.574845679

Combine like terms: -1 + 4.574845679 = 3.574845679
4.574845679 + -4.277777778x + x2 = 3.574845679

Factor a perfect square on the left side:
(x + -2.138888889)(x + -2.138888889) = 3.574845679

Calculate the square root of the right side: 1.890726231

Break this problem into two subproblems by setting 
(x + -2.138888889) equal to 1.890726231 and -1.890726231.

Subproblem 1

x + -2.138888889 = 1.890726231 Simplifying x + -2.138888889 = 1.890726231 Reorder the terms: -2.138888889 + x = 1.890726231 Solving -2.138888889 + x = 1.890726231 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.138888889' to each side of the equation. -2.138888889 + 2.138888889 + x = 1.890726231 + 2.138888889 Combine like terms: -2.138888889 + 2.138888889 = 0.000000000 0.000000000 + x = 1.890726231 + 2.138888889 x = 1.890726231 + 2.138888889 Combine like terms: 1.890726231 + 2.138888889 = 4.02961512 x = 4.02961512 Simplifying x = 4.02961512

Subproblem 2

x + -2.138888889 = -1.890726231 Simplifying x + -2.138888889 = -1.890726231 Reorder the terms: -2.138888889 + x = -1.890726231 Solving -2.138888889 + x = -1.890726231 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.138888889' to each side of the equation. -2.138888889 + 2.138888889 + x = -1.890726231 + 2.138888889 Combine like terms: -2.138888889 + 2.138888889 = 0.000000000 0.000000000 + x = -1.890726231 + 2.138888889 x = -1.890726231 + 2.138888889 Combine like terms: -1.890726231 + 2.138888889 = 0.248162658 x = 0.248162658 Simplifying x = 0.248162658

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.02961512, 0.248162658}

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