7x^2-51x-14=0

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Solution for 7x^2-51x-14=0 equation:


Simplifying
7x2 + -51x + -14 = 0

Reorder the terms:
-14 + -51x + 7x2 = 0

Solving
-14 + -51x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
-2 + -7.285714286x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -7.285714286x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -7.285714286x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -7.285714286x + x2 = 0 + 2
-7.285714286x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-7.285714286x + x2 = 2

The x term is -7.285714286x.  Take half its coefficient (-3.642857143).
Square it (13.27040816) and add it to both sides.

Add '13.27040816' to each side of the equation.
-7.285714286x + 13.27040816 + x2 = 2 + 13.27040816

Reorder the terms:
13.27040816 + -7.285714286x + x2 = 2 + 13.27040816

Combine like terms: 2 + 13.27040816 = 15.27040816
13.27040816 + -7.285714286x + x2 = 15.27040816

Factor a perfect square on the left side:
(x + -3.642857143)(x + -3.642857143) = 15.27040816

Calculate the square root of the right side: 3.907736961

Break this problem into two subproblems by setting 
(x + -3.642857143) equal to 3.907736961 and -3.907736961.

Subproblem 1

x + -3.642857143 = 3.907736961 Simplifying x + -3.642857143 = 3.907736961 Reorder the terms: -3.642857143 + x = 3.907736961 Solving -3.642857143 + x = 3.907736961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.642857143' to each side of the equation. -3.642857143 + 3.642857143 + x = 3.907736961 + 3.642857143 Combine like terms: -3.642857143 + 3.642857143 = 0.000000000 0.000000000 + x = 3.907736961 + 3.642857143 x = 3.907736961 + 3.642857143 Combine like terms: 3.907736961 + 3.642857143 = 7.550594104 x = 7.550594104 Simplifying x = 7.550594104

Subproblem 2

x + -3.642857143 = -3.907736961 Simplifying x + -3.642857143 = -3.907736961 Reorder the terms: -3.642857143 + x = -3.907736961 Solving -3.642857143 + x = -3.907736961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.642857143' to each side of the equation. -3.642857143 + 3.642857143 + x = -3.907736961 + 3.642857143 Combine like terms: -3.642857143 + 3.642857143 = 0.000000000 0.000000000 + x = -3.907736961 + 3.642857143 x = -3.907736961 + 3.642857143 Combine like terms: -3.907736961 + 3.642857143 = -0.264879818 x = -0.264879818 Simplifying x = -0.264879818

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.550594104, -0.264879818}

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