345x*x-6788x=678

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Solution for 345x*x-6788x=678 equation:


Simplifying
345x * x + -6788x = 678

Multiply x * x
345x2 + -6788x = 678

Reorder the terms:
-6788x + 345x2 = 678

Solving
-6788x + 345x2 = 678

Solving for variable 'x'.

Reorder the terms:
-678 + -6788x + 345x2 = 678 + -678

Combine like terms: 678 + -678 = 0
-678 + -6788x + 345x2 = 0

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

Divide each side by '345'.
-1.965217391 + -19.67536232x + x2 = 0

Move the constant term to the right:

Add '1.965217391' to each side of the equation.
-1.965217391 + -19.67536232x + 1.965217391 + x2 = 0 + 1.965217391

Reorder the terms:
-1.965217391 + 1.965217391 + -19.67536232x + x2 = 0 + 1.965217391

Combine like terms: -1.965217391 + 1.965217391 = 0.000000000
0.000000000 + -19.67536232x + x2 = 0 + 1.965217391
-19.67536232x + x2 = 0 + 1.965217391

Combine like terms: 0 + 1.965217391 = 1.965217391
-19.67536232x + x2 = 1.965217391

The x term is -19.67536232x.  Take half its coefficient (-9.83768116).
Square it (96.77997061) and add it to both sides.

Add '96.77997061' to each side of the equation.
-19.67536232x + 96.77997061 + x2 = 1.965217391 + 96.77997061

Reorder the terms:
96.77997061 + -19.67536232x + x2 = 1.965217391 + 96.77997061

Combine like terms: 1.965217391 + 96.77997061 = 98.745188001
96.77997061 + -19.67536232x + x2 = 98.745188001

Factor a perfect square on the left side:
(x + -9.83768116)(x + -9.83768116) = 98.745188001

Calculate the square root of the right side: 9.937061336

Break this problem into two subproblems by setting 
(x + -9.83768116) equal to 9.937061336 and -9.937061336.

Subproblem 1

x + -9.83768116 = 9.937061336 Simplifying x + -9.83768116 = 9.937061336 Reorder the terms: -9.83768116 + x = 9.937061336 Solving -9.83768116 + x = 9.937061336 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.83768116' to each side of the equation. -9.83768116 + 9.83768116 + x = 9.937061336 + 9.83768116 Combine like terms: -9.83768116 + 9.83768116 = 0.00000000 0.00000000 + x = 9.937061336 + 9.83768116 x = 9.937061336 + 9.83768116 Combine like terms: 9.937061336 + 9.83768116 = 19.774742496 x = 19.774742496 Simplifying x = 19.774742496

Subproblem 2

x + -9.83768116 = -9.937061336 Simplifying x + -9.83768116 = -9.937061336 Reorder the terms: -9.83768116 + x = -9.937061336 Solving -9.83768116 + x = -9.937061336 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.83768116' to each side of the equation. -9.83768116 + 9.83768116 + x = -9.937061336 + 9.83768116 Combine like terms: -9.83768116 + 9.83768116 = 0.00000000 0.00000000 + x = -9.937061336 + 9.83768116 x = -9.937061336 + 9.83768116 Combine like terms: -9.937061336 + 9.83768116 = -0.099380176 x = -0.099380176 Simplifying x = -0.099380176

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.774742496, -0.099380176}

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