2x^2+14x=1

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Solution for 2x^2+14x=1 equation:


Simplifying
2x2 + 14x = 1

Reorder the terms:
14x + 2x2 = 1

Solving
14x + 2x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + 14x + 2x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 14x + 2x2 = 0

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

Divide each side by '2'.
-0.5 + 7x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 7x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 7x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 7x + x2 = 0 + 0.5
7x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
7x + x2 = 0.5

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = 0.5 + 12.25

Reorder the terms:
12.25 + 7x + x2 = 0.5 + 12.25

Combine like terms: 0.5 + 12.25 = 12.75
12.25 + 7x + x2 = 12.75

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 12.75

Calculate the square root of the right side: 3.570714214

Break this problem into two subproblems by setting 
(x + 3.5) equal to 3.570714214 and -3.570714214.

Subproblem 1

x + 3.5 = 3.570714214 Simplifying x + 3.5 = 3.570714214 Reorder the terms: 3.5 + x = 3.570714214 Solving 3.5 + x = 3.570714214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 3.570714214 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 3.570714214 + -3.5 x = 3.570714214 + -3.5 Combine like terms: 3.570714214 + -3.5 = 0.070714214 x = 0.070714214 Simplifying x = 0.070714214

Subproblem 2

x + 3.5 = -3.570714214 Simplifying x + 3.5 = -3.570714214 Reorder the terms: 3.5 + x = -3.570714214 Solving 3.5 + x = -3.570714214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -3.570714214 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -3.570714214 + -3.5 x = -3.570714214 + -3.5 Combine like terms: -3.570714214 + -3.5 = -7.070714214 x = -7.070714214 Simplifying x = -7.070714214

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.070714214, -7.070714214}

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