3x^2-78x+500=0

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Solution for 3x^2-78x+500=0 equation:


Simplifying
3x2 + -78x + 500 = 0

Reorder the terms:
500 + -78x + 3x2 = 0

Solving
500 + -78x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
166.6666667 + -26x + x2 = 0

Move the constant term to the right:

Add '-166.6666667' to each side of the equation.
166.6666667 + -26x + -166.6666667 + x2 = 0 + -166.6666667

Reorder the terms:
166.6666667 + -166.6666667 + -26x + x2 = 0 + -166.6666667

Combine like terms: 166.6666667 + -166.6666667 = 0.0000000
0.0000000 + -26x + x2 = 0 + -166.6666667
-26x + x2 = 0 + -166.6666667

Combine like terms: 0 + -166.6666667 = -166.6666667
-26x + x2 = -166.6666667

The x term is -26x.  Take half its coefficient (-13).
Square it (169) and add it to both sides.

Add '169' to each side of the equation.
-26x + 169 + x2 = -166.6666667 + 169

Reorder the terms:
169 + -26x + x2 = -166.6666667 + 169

Combine like terms: -166.6666667 + 169 = 2.3333333
169 + -26x + x2 = 2.3333333

Factor a perfect square on the left side:
(x + -13)(x + -13) = 2.3333333

Calculate the square root of the right side: 1.527525221

Break this problem into two subproblems by setting 
(x + -13) equal to 1.527525221 and -1.527525221.

Subproblem 1

x + -13 = 1.527525221 Simplifying x + -13 = 1.527525221 Reorder the terms: -13 + x = 1.527525221 Solving -13 + x = 1.527525221 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13' to each side of the equation. -13 + 13 + x = 1.527525221 + 13 Combine like terms: -13 + 13 = 0 0 + x = 1.527525221 + 13 x = 1.527525221 + 13 Combine like terms: 1.527525221 + 13 = 14.527525221 x = 14.527525221 Simplifying x = 14.527525221

Subproblem 2

x + -13 = -1.527525221 Simplifying x + -13 = -1.527525221 Reorder the terms: -13 + x = -1.527525221 Solving -13 + x = -1.527525221 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13' to each side of the equation. -13 + 13 + x = -1.527525221 + 13 Combine like terms: -13 + 13 = 0 0 + x = -1.527525221 + 13 x = -1.527525221 + 13 Combine like terms: -1.527525221 + 13 = 11.472474779 x = 11.472474779 Simplifying x = 11.472474779

Solution

The solution to the problem is based on the solutions from the subproblems. x = {14.527525221, 11.472474779}

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