4x^2+x=196

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Solution for 4x^2+x=196 equation:


Simplifying
4x2 + x = 196

Reorder the terms:
x + 4x2 = 196

Solving
x + 4x2 = 196

Solving for variable 'x'.

Reorder the terms:
-196 + x + 4x2 = 196 + -196

Combine like terms: 196 + -196 = 0
-196 + x + 4x2 = 0

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

Divide each side by '4'.
-49 + 0.25x + x2 = 0

Move the constant term to the right:

Add '49' to each side of the equation.
-49 + 0.25x + 49 + x2 = 0 + 49

Reorder the terms:
-49 + 49 + 0.25x + x2 = 0 + 49

Combine like terms: -49 + 49 = 0
0 + 0.25x + x2 = 0 + 49
0.25x + x2 = 0 + 49

Combine like terms: 0 + 49 = 49
0.25x + x2 = 49

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.25x + 0.25 + x2 = 49 + 0.25

Reorder the terms:
0.25 + 0.25x + x2 = 49 + 0.25

Combine like terms: 49 + 0.25 = 49.25
0.25 + 0.25x + x2 = 49.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 49.25

Calculate the square root of the right side: 7.017834424

Break this problem into two subproblems by setting 
(x + 0.5) equal to 7.017834424 and -7.017834424.

Subproblem 1

x + 0.5 = 7.017834424 Simplifying x + 0.5 = 7.017834424 Reorder the terms: 0.5 + x = 7.017834424 Solving 0.5 + x = 7.017834424 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 7.017834424 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.017834424 + -0.5 x = 7.017834424 + -0.5 Combine like terms: 7.017834424 + -0.5 = 6.517834424 x = 6.517834424 Simplifying x = 6.517834424

Subproblem 2

x + 0.5 = -7.017834424 Simplifying x + 0.5 = -7.017834424 Reorder the terms: 0.5 + x = -7.017834424 Solving 0.5 + x = -7.017834424 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -7.017834424 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.017834424 + -0.5 x = -7.017834424 + -0.5 Combine like terms: -7.017834424 + -0.5 = -7.517834424 x = -7.517834424 Simplifying x = -7.517834424

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.517834424, -7.517834424}

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