7x^2=37x+5

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Solution for 7x^2=37x+5 equation:


Simplifying
7x2 = 37x + 5

Reorder the terms:
7x2 = 5 + 37x

Solving
7x2 = 5 + 37x

Solving for variable 'x'.

Reorder the terms:
-5 + -37x + 7x2 = 5 + 37x + -5 + -37x

Reorder the terms:
-5 + -37x + 7x2 = 5 + -5 + 37x + -37x

Combine like terms: 5 + -5 = 0
-5 + -37x + 7x2 = 0 + 37x + -37x
-5 + -37x + 7x2 = 37x + -37x

Combine like terms: 37x + -37x = 0
-5 + -37x + 7x2 = 0

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

Divide each side by '7'.
-0.7142857143 + -5.285714286x + x2 = 0

Move the constant term to the right:

Add '0.7142857143' to each side of the equation.
-0.7142857143 + -5.285714286x + 0.7142857143 + x2 = 0 + 0.7142857143

Reorder the terms:
-0.7142857143 + 0.7142857143 + -5.285714286x + x2 = 0 + 0.7142857143

Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000
0.0000000000 + -5.285714286x + x2 = 0 + 0.7142857143
-5.285714286x + x2 = 0 + 0.7142857143

Combine like terms: 0 + 0.7142857143 = 0.7142857143
-5.285714286x + x2 = 0.7142857143

The x term is -5.285714286x.  Take half its coefficient (-2.642857143).
Square it (6.984693878) and add it to both sides.

Add '6.984693878' to each side of the equation.
-5.285714286x + 6.984693878 + x2 = 0.7142857143 + 6.984693878

Reorder the terms:
6.984693878 + -5.285714286x + x2 = 0.7142857143 + 6.984693878

Combine like terms: 0.7142857143 + 6.984693878 = 7.6989795923
6.984693878 + -5.285714286x + x2 = 7.6989795923

Factor a perfect square on the left side:
(x + -2.642857143)(x + -2.642857143) = 7.6989795923

Calculate the square root of the right side: 2.774703514

Break this problem into two subproblems by setting 
(x + -2.642857143) equal to 2.774703514 and -2.774703514.

Subproblem 1

x + -2.642857143 = 2.774703514 Simplifying x + -2.642857143 = 2.774703514 Reorder the terms: -2.642857143 + x = 2.774703514 Solving -2.642857143 + x = 2.774703514 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.642857143' to each side of the equation. -2.642857143 + 2.642857143 + x = 2.774703514 + 2.642857143 Combine like terms: -2.642857143 + 2.642857143 = 0.000000000 0.000000000 + x = 2.774703514 + 2.642857143 x = 2.774703514 + 2.642857143 Combine like terms: 2.774703514 + 2.642857143 = 5.417560657 x = 5.417560657 Simplifying x = 5.417560657

Subproblem 2

x + -2.642857143 = -2.774703514 Simplifying x + -2.642857143 = -2.774703514 Reorder the terms: -2.642857143 + x = -2.774703514 Solving -2.642857143 + x = -2.774703514 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.642857143' to each side of the equation. -2.642857143 + 2.642857143 + x = -2.774703514 + 2.642857143 Combine like terms: -2.642857143 + 2.642857143 = 0.000000000 0.000000000 + x = -2.774703514 + 2.642857143 x = -2.774703514 + 2.642857143 Combine like terms: -2.774703514 + 2.642857143 = -0.131846371 x = -0.131846371 Simplifying x = -0.131846371

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.417560657, -0.131846371}

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