6x^2-78x+175=0

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Solution for 6x^2-78x+175=0 equation:


Simplifying
6x2 + -78x + 175 = 0

Reorder the terms:
175 + -78x + 6x2 = 0

Solving
175 + -78x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
29.16666667 + -13x + x2 = 0

Move the constant term to the right:

Add '-29.16666667' to each side of the equation.
29.16666667 + -13x + -29.16666667 + x2 = 0 + -29.16666667

Reorder the terms:
29.16666667 + -29.16666667 + -13x + x2 = 0 + -29.16666667

Combine like terms: 29.16666667 + -29.16666667 = 0.00000000
0.00000000 + -13x + x2 = 0 + -29.16666667
-13x + x2 = 0 + -29.16666667

Combine like terms: 0 + -29.16666667 = -29.16666667
-13x + x2 = -29.16666667

The x term is -13x.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13x + 42.25 + x2 = -29.16666667 + 42.25

Reorder the terms:
42.25 + -13x + x2 = -29.16666667 + 42.25

Combine like terms: -29.16666667 + 42.25 = 13.08333333
42.25 + -13x + x2 = 13.08333333

Factor a perfect square on the left side:
(x + -6.5)(x + -6.5) = 13.08333333

Calculate the square root of the right side: 3.617089069

Break this problem into two subproblems by setting 
(x + -6.5) equal to 3.617089069 and -3.617089069.

Subproblem 1

x + -6.5 = 3.617089069 Simplifying x + -6.5 = 3.617089069 Reorder the terms: -6.5 + x = 3.617089069 Solving -6.5 + x = 3.617089069 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x = 3.617089069 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x = 3.617089069 + 6.5 x = 3.617089069 + 6.5 Combine like terms: 3.617089069 + 6.5 = 10.117089069 x = 10.117089069 Simplifying x = 10.117089069

Subproblem 2

x + -6.5 = -3.617089069 Simplifying x + -6.5 = -3.617089069 Reorder the terms: -6.5 + x = -3.617089069 Solving -6.5 + x = -3.617089069 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x = -3.617089069 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x = -3.617089069 + 6.5 x = -3.617089069 + 6.5 Combine like terms: -3.617089069 + 6.5 = 2.882910931 x = 2.882910931 Simplifying x = 2.882910931

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10.117089069, 2.882910931}

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