6.75x^2-21.8x+16=0

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Solution for 6.75x^2-21.8x+16=0 equation:


Simplifying
6.75x2 + -21.8x + 16 = 0

Reorder the terms:
16 + -21.8x + 6.75x2 = 0

Solving
16 + -21.8x + 6.75x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
6.75 the coefficient of the squared term: 

Divide each side by '6.75'.
2.37037037 + -3.22962963x + x2 = 0

Move the constant term to the right:

Add '-2.37037037' to each side of the equation.
2.37037037 + -3.22962963x + -2.37037037 + x2 = 0 + -2.37037037

Reorder the terms:
2.37037037 + -2.37037037 + -3.22962963x + x2 = 0 + -2.37037037

Combine like terms: 2.37037037 + -2.37037037 = 0.00000000
0.00000000 + -3.22962963x + x2 = 0 + -2.37037037
-3.22962963x + x2 = 0 + -2.37037037

Combine like terms: 0 + -2.37037037 = -2.37037037
-3.22962963x + x2 = -2.37037037

The x term is -3.22962963x.  Take half its coefficient (-1.614814815).
Square it (2.607626887) and add it to both sides.

Add '2.607626887' to each side of the equation.
-3.22962963x + 2.607626887 + x2 = -2.37037037 + 2.607626887

Reorder the terms:
2.607626887 + -3.22962963x + x2 = -2.37037037 + 2.607626887

Combine like terms: -2.37037037 + 2.607626887 = 0.237256517
2.607626887 + -3.22962963x + x2 = 0.237256517

Factor a perfect square on the left side:
(x + -1.614814815)(x + -1.614814815) = 0.237256517

Calculate the square root of the right side: 0.487089845

Break this problem into two subproblems by setting 
(x + -1.614814815) equal to 0.487089845 and -0.487089845.

Subproblem 1

x + -1.614814815 = 0.487089845 Simplifying x + -1.614814815 = 0.487089845 Reorder the terms: -1.614814815 + x = 0.487089845 Solving -1.614814815 + x = 0.487089845 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.614814815' to each side of the equation. -1.614814815 + 1.614814815 + x = 0.487089845 + 1.614814815 Combine like terms: -1.614814815 + 1.614814815 = 0.000000000 0.000000000 + x = 0.487089845 + 1.614814815 x = 0.487089845 + 1.614814815 Combine like terms: 0.487089845 + 1.614814815 = 2.10190466 x = 2.10190466 Simplifying x = 2.10190466

Subproblem 2

x + -1.614814815 = -0.487089845 Simplifying x + -1.614814815 = -0.487089845 Reorder the terms: -1.614814815 + x = -0.487089845 Solving -1.614814815 + x = -0.487089845 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.614814815' to each side of the equation. -1.614814815 + 1.614814815 + x = -0.487089845 + 1.614814815 Combine like terms: -1.614814815 + 1.614814815 = 0.000000000 0.000000000 + x = -0.487089845 + 1.614814815 x = -0.487089845 + 1.614814815 Combine like terms: -0.487089845 + 1.614814815 = 1.12772497 x = 1.12772497 Simplifying x = 1.12772497

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.10190466, 1.12772497}

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