6.59=x^2+3x+1

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Solution for 6.59=x^2+3x+1 equation:


Simplifying
6.59 = x2 + 3x + 1

Reorder the terms:
6.59 = 1 + 3x + x2

Solving
6.59 = 1 + 3x + x2

Solving for variable 'x'.

Combine like terms: 6.59 + -1 = 5.59
5.59 + -3x + -1x2 = 1 + 3x + x2 + -1 + -3x + -1x2

Reorder the terms:
5.59 + -3x + -1x2 = 1 + -1 + 3x + -3x + x2 + -1x2

Combine like terms: 1 + -1 = 0
5.59 + -3x + -1x2 = 0 + 3x + -3x + x2 + -1x2
5.59 + -3x + -1x2 = 3x + -3x + x2 + -1x2

Combine like terms: 3x + -3x = 0
5.59 + -3x + -1x2 = 0 + x2 + -1x2
5.59 + -3x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
5.59 + -3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-5.59 + 3x + x2 = 0

Move the constant term to the right:

Add '5.59' to each side of the equation.
-5.59 + 3x + 5.59 + x2 = 0 + 5.59

Reorder the terms:
-5.59 + 5.59 + 3x + x2 = 0 + 5.59

Combine like terms: -5.59 + 5.59 = 0.00
0.00 + 3x + x2 = 0 + 5.59
3x + x2 = 0 + 5.59

Combine like terms: 0 + 5.59 = 5.59
3x + x2 = 5.59

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 5.59 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 5.59 + 2.25

Combine like terms: 5.59 + 2.25 = 7.84
2.25 + 3x + x2 = 7.84

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 7.84

Calculate the square root of the right side: 2.8

Break this problem into two subproblems by setting 
(x + 1.5) equal to 2.8 and -2.8.

Subproblem 1

x + 1.5 = 2.8 Simplifying x + 1.5 = 2.8 Reorder the terms: 1.5 + x = 2.8 Solving 1.5 + x = 2.8 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 2.8 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 2.8 + -1.5 x = 2.8 + -1.5 Combine like terms: 2.8 + -1.5 = 1.3 x = 1.3 Simplifying x = 1.3

Subproblem 2

x + 1.5 = -2.8 Simplifying x + 1.5 = -2.8 Reorder the terms: 1.5 + x = -2.8 Solving 1.5 + x = -2.8 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -2.8 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -2.8 + -1.5 x = -2.8 + -1.5 Combine like terms: -2.8 + -1.5 = -4.3 x = -4.3 Simplifying x = -4.3

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3, -4.3}

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