5g^2-2g-5=0

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Solution for 5g^2-2g-5=0 equation:


Simplifying
5g2 + -2g + -5 = 0

Reorder the terms:
-5 + -2g + 5g2 = 0

Solving
-5 + -2g + 5g2 = 0

Solving for variable 'g'.

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

Divide each side by '5'.
-1 + -0.4g + g2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.4g + 1 + g2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.4g + g2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.4g + g2 = 0 + 1
-0.4g + g2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.4g + g2 = 1

The g term is -0.4g.  Take half its coefficient (-0.2).
Square it (0.04) and add it to both sides.

Add '0.04' to each side of the equation.
-0.4g + 0.04 + g2 = 1 + 0.04

Reorder the terms:
0.04 + -0.4g + g2 = 1 + 0.04

Combine like terms: 1 + 0.04 = 1.04
0.04 + -0.4g + g2 = 1.04

Factor a perfect square on the left side:
(g + -0.2)(g + -0.2) = 1.04

Calculate the square root of the right side: 1.019803903

Break this problem into two subproblems by setting 
(g + -0.2) equal to 1.019803903 and -1.019803903.

Subproblem 1

g + -0.2 = 1.019803903 Simplifying g + -0.2 = 1.019803903 Reorder the terms: -0.2 + g = 1.019803903 Solving -0.2 + g = 1.019803903 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + g = 1.019803903 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + g = 1.019803903 + 0.2 g = 1.019803903 + 0.2 Combine like terms: 1.019803903 + 0.2 = 1.219803903 g = 1.219803903 Simplifying g = 1.219803903

Subproblem 2

g + -0.2 = -1.019803903 Simplifying g + -0.2 = -1.019803903 Reorder the terms: -0.2 + g = -1.019803903 Solving -0.2 + g = -1.019803903 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + g = -1.019803903 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + g = -1.019803903 + 0.2 g = -1.019803903 + 0.2 Combine like terms: -1.019803903 + 0.2 = -0.819803903 g = -0.819803903 Simplifying g = -0.819803903

Solution

The solution to the problem is based on the solutions from the subproblems. g = {1.219803903, -0.819803903}

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