x*x+4x+1=32

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Solution for x*x+4x+1=32 equation:


Simplifying
x * x + 4x + 1 = 32

Multiply x * x
x2 + 4x + 1 = 32

Reorder the terms:
1 + 4x + x2 = 32

Solving
1 + 4x + x2 = 32

Solving for variable 'x'.

Reorder the terms:
1 + -32 + 4x + x2 = 32 + -32

Combine like terms: 1 + -32 = -31
-31 + 4x + x2 = 32 + -32

Combine like terms: 32 + -32 = 0
-31 + 4x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + 4x + 31 + x2 = 0 + 31

Reorder the terms:
-31 + 31 + 4x + x2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + 4x + x2 = 0 + 31
4x + x2 = 0 + 31

Combine like terms: 0 + 31 = 31
4x + x2 = 31

The x term is 4x.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4x + 4 + x2 = 31 + 4

Reorder the terms:
4 + 4x + x2 = 31 + 4

Combine like terms: 31 + 4 = 35
4 + 4x + x2 = 35

Factor a perfect square on the left side:
(x + 2)(x + 2) = 35

Calculate the square root of the right side: 5.916079783

Break this problem into two subproblems by setting 
(x + 2) equal to 5.916079783 and -5.916079783.

Subproblem 1

x + 2 = 5.916079783 Simplifying x + 2 = 5.916079783 Reorder the terms: 2 + x = 5.916079783 Solving 2 + x = 5.916079783 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = 5.916079783 + -2 Combine like terms: 2 + -2 = 0 0 + x = 5.916079783 + -2 x = 5.916079783 + -2 Combine like terms: 5.916079783 + -2 = 3.916079783 x = 3.916079783 Simplifying x = 3.916079783

Subproblem 2

x + 2 = -5.916079783 Simplifying x + 2 = -5.916079783 Reorder the terms: 2 + x = -5.916079783 Solving 2 + x = -5.916079783 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = -5.916079783 + -2 Combine like terms: 2 + -2 = 0 0 + x = -5.916079783 + -2 x = -5.916079783 + -2 Combine like terms: -5.916079783 + -2 = -7.916079783 x = -7.916079783 Simplifying x = -7.916079783

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.916079783, -7.916079783}

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