45x^2+57x-2314=0

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Solution for 45x^2+57x-2314=0 equation:


Simplifying
45x2 + 57x + -2314 = 0

Reorder the terms:
-2314 + 57x + 45x2 = 0

Solving
-2314 + 57x + 45x2 = 0

Solving for variable 'x'.

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

Divide each side by '45'.
-51.42222222 + 1.266666667x + x2 = 0

Move the constant term to the right:

Add '51.42222222' to each side of the equation.
-51.42222222 + 1.266666667x + 51.42222222 + x2 = 0 + 51.42222222

Reorder the terms:
-51.42222222 + 51.42222222 + 1.266666667x + x2 = 0 + 51.42222222

Combine like terms: -51.42222222 + 51.42222222 = 0.00000000
0.00000000 + 1.266666667x + x2 = 0 + 51.42222222
1.266666667x + x2 = 0 + 51.42222222

Combine like terms: 0 + 51.42222222 = 51.42222222
1.266666667x + x2 = 51.42222222

The x term is 1.266666667x.  Take half its coefficient (0.6333333335).
Square it (0.4011111113) and add it to both sides.

Add '0.4011111113' to each side of the equation.
1.266666667x + 0.4011111113 + x2 = 51.42222222 + 0.4011111113

Reorder the terms:
0.4011111113 + 1.266666667x + x2 = 51.42222222 + 0.4011111113

Combine like terms: 51.42222222 + 0.4011111113 = 51.8233333313
0.4011111113 + 1.266666667x + x2 = 51.8233333313

Factor a perfect square on the left side:
(x + 0.6333333335)(x + 0.6333333335) = 51.8233333313

Calculate the square root of the right side: 7.198842499

Break this problem into two subproblems by setting 
(x + 0.6333333335) equal to 7.198842499 and -7.198842499.

Subproblem 1

x + 0.6333333335 = 7.198842499 Simplifying x + 0.6333333335 = 7.198842499 Reorder the terms: 0.6333333335 + x = 7.198842499 Solving 0.6333333335 + x = 7.198842499 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6333333335' to each side of the equation. 0.6333333335 + -0.6333333335 + x = 7.198842499 + -0.6333333335 Combine like terms: 0.6333333335 + -0.6333333335 = 0.0000000000 0.0000000000 + x = 7.198842499 + -0.6333333335 x = 7.198842499 + -0.6333333335 Combine like terms: 7.198842499 + -0.6333333335 = 6.5655091655 x = 6.5655091655 Simplifying x = 6.5655091655

Subproblem 2

x + 0.6333333335 = -7.198842499 Simplifying x + 0.6333333335 = -7.198842499 Reorder the terms: 0.6333333335 + x = -7.198842499 Solving 0.6333333335 + x = -7.198842499 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6333333335' to each side of the equation. 0.6333333335 + -0.6333333335 + x = -7.198842499 + -0.6333333335 Combine like terms: 0.6333333335 + -0.6333333335 = 0.0000000000 0.0000000000 + x = -7.198842499 + -0.6333333335 x = -7.198842499 + -0.6333333335 Combine like terms: -7.198842499 + -0.6333333335 = -7.8321758325 x = -7.8321758325 Simplifying x = -7.8321758325

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.5655091655, -7.8321758325}

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