10x^2+31x+5=0

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Solution for 10x^2+31x+5=0 equation:


Simplifying
10x2 + 31x + 5 = 0

Reorder the terms:
5 + 31x + 10x2 = 0

Solving
5 + 31x + 10x2 = 0

Solving for variable 'x'.

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

Divide each side by '10'.
0.5 + 3.1x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 3.1x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 3.1x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 3.1x + x2 = 0 + -0.5
3.1x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
3.1x + x2 = -0.5

The x term is 3.1x.  Take half its coefficient (1.55).
Square it (2.4025) and add it to both sides.

Add '2.4025' to each side of the equation.
3.1x + 2.4025 + x2 = -0.5 + 2.4025

Reorder the terms:
2.4025 + 3.1x + x2 = -0.5 + 2.4025

Combine like terms: -0.5 + 2.4025 = 1.9025
2.4025 + 3.1x + x2 = 1.9025

Factor a perfect square on the left side:
(x + 1.55)(x + 1.55) = 1.9025

Calculate the square root of the right side: 1.379311422

Break this problem into two subproblems by setting 
(x + 1.55) equal to 1.379311422 and -1.379311422.

Subproblem 1

x + 1.55 = 1.379311422 Simplifying x + 1.55 = 1.379311422 Reorder the terms: 1.55 + x = 1.379311422 Solving 1.55 + x = 1.379311422 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.55' to each side of the equation. 1.55 + -1.55 + x = 1.379311422 + -1.55 Combine like terms: 1.55 + -1.55 = 0.00 0.00 + x = 1.379311422 + -1.55 x = 1.379311422 + -1.55 Combine like terms: 1.379311422 + -1.55 = -0.170688578 x = -0.170688578 Simplifying x = -0.170688578

Subproblem 2

x + 1.55 = -1.379311422 Simplifying x + 1.55 = -1.379311422 Reorder the terms: 1.55 + x = -1.379311422 Solving 1.55 + x = -1.379311422 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.55' to each side of the equation. 1.55 + -1.55 + x = -1.379311422 + -1.55 Combine like terms: 1.55 + -1.55 = 0.00 0.00 + x = -1.379311422 + -1.55 x = -1.379311422 + -1.55 Combine like terms: -1.379311422 + -1.55 = -2.929311422 x = -2.929311422 Simplifying x = -2.929311422

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.170688578, -2.929311422}

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