x^2+4x^2+1x+1=461

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Solution for x^2+4x^2+1x+1=461 equation:


Simplifying
x2 + 4x2 + 1x + 1 = 461

Reorder the terms:
1 + 1x + x2 + 4x2 = 461

Combine like terms: x2 + 4x2 = 5x2
1 + 1x + 5x2 = 461

Solving
1 + 1x + 5x2 = 461

Solving for variable 'x'.

Reorder the terms:
1 + -461 + 1x + 5x2 = 461 + -461

Combine like terms: 1 + -461 = -460
-460 + 1x + 5x2 = 461 + -461

Combine like terms: 461 + -461 = 0
-460 + 1x + 5x2 = 0

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

Divide each side by '5'.
-92 + 0.2x + x2 = 0

Move the constant term to the right:

Add '92' to each side of the equation.
-92 + 0.2x + 92 + x2 = 0 + 92

Reorder the terms:
-92 + 92 + 0.2x + x2 = 0 + 92

Combine like terms: -92 + 92 = 0
0 + 0.2x + x2 = 0 + 92
0.2x + x2 = 0 + 92

Combine like terms: 0 + 92 = 92
0.2x + x2 = 92

The x term is 0.2x.  Take half its coefficient (0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
0.2x + 0.01 + x2 = 92 + 0.01

Reorder the terms:
0.01 + 0.2x + x2 = 92 + 0.01

Combine like terms: 92 + 0.01 = 92.01
0.01 + 0.2x + x2 = 92.01

Factor a perfect square on the left side:
(x + 0.1)(x + 0.1) = 92.01

Calculate the square root of the right side: 9.592184318

Break this problem into two subproblems by setting 
(x + 0.1) equal to 9.592184318 and -9.592184318.

Subproblem 1

x + 0.1 = 9.592184318 Simplifying x + 0.1 = 9.592184318 Reorder the terms: 0.1 + x = 9.592184318 Solving 0.1 + x = 9.592184318 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = 9.592184318 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = 9.592184318 + -0.1 x = 9.592184318 + -0.1 Combine like terms: 9.592184318 + -0.1 = 9.492184318 x = 9.492184318 Simplifying x = 9.492184318

Subproblem 2

x + 0.1 = -9.592184318 Simplifying x + 0.1 = -9.592184318 Reorder the terms: 0.1 + x = -9.592184318 Solving 0.1 + x = -9.592184318 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = -9.592184318 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = -9.592184318 + -0.1 x = -9.592184318 + -0.1 Combine like terms: -9.592184318 + -0.1 = -9.692184318 x = -9.692184318 Simplifying x = -9.692184318

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.492184318, -9.692184318}

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