14x^2+53x=14

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Solution for 14x^2+53x=14 equation:


Simplifying
14x2 + 53x = 14

Reorder the terms:
53x + 14x2 = 14

Solving
53x + 14x2 = 14

Solving for variable 'x'.

Reorder the terms:
-14 + 53x + 14x2 = 14 + -14

Combine like terms: 14 + -14 = 0
-14 + 53x + 14x2 = 0

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

Divide each side by '14'.
-1 + 3.785714286x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 3.785714286x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 3.785714286x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 3.785714286x + x2 = 0 + 1
3.785714286x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
3.785714286x + x2 = 1

The x term is 3.785714286x.  Take half its coefficient (1.892857143).
Square it (3.582908164) and add it to both sides.

Add '3.582908164' to each side of the equation.
3.785714286x + 3.582908164 + x2 = 1 + 3.582908164

Reorder the terms:
3.582908164 + 3.785714286x + x2 = 1 + 3.582908164

Combine like terms: 1 + 3.582908164 = 4.582908164
3.582908164 + 3.785714286x + x2 = 4.582908164

Factor a perfect square on the left side:
(x + 1.892857143)(x + 1.892857143) = 4.582908164

Calculate the square root of the right side: 2.140772796

Break this problem into two subproblems by setting 
(x + 1.892857143) equal to 2.140772796 and -2.140772796.

Subproblem 1

x + 1.892857143 = 2.140772796 Simplifying x + 1.892857143 = 2.140772796 Reorder the terms: 1.892857143 + x = 2.140772796 Solving 1.892857143 + x = 2.140772796 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.892857143' to each side of the equation. 1.892857143 + -1.892857143 + x = 2.140772796 + -1.892857143 Combine like terms: 1.892857143 + -1.892857143 = 0.000000000 0.000000000 + x = 2.140772796 + -1.892857143 x = 2.140772796 + -1.892857143 Combine like terms: 2.140772796 + -1.892857143 = 0.247915653 x = 0.247915653 Simplifying x = 0.247915653

Subproblem 2

x + 1.892857143 = -2.140772796 Simplifying x + 1.892857143 = -2.140772796 Reorder the terms: 1.892857143 + x = -2.140772796 Solving 1.892857143 + x = -2.140772796 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.892857143' to each side of the equation. 1.892857143 + -1.892857143 + x = -2.140772796 + -1.892857143 Combine like terms: 1.892857143 + -1.892857143 = 0.000000000 0.000000000 + x = -2.140772796 + -1.892857143 x = -2.140772796 + -1.892857143 Combine like terms: -2.140772796 + -1.892857143 = -4.033629939 x = -4.033629939 Simplifying x = -4.033629939

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.247915653, -4.033629939}

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