y^2+10y+3=0

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Solution for y^2+10y+3=0 equation:


Simplifying
y2 + 10y + 3 = 0

Reorder the terms:
3 + 10y + y2 = 0

Solving
3 + 10y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 10y + -3 + y2 = 0 + -3

Reorder the terms:
3 + -3 + 10y + y2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 10y + y2 = 0 + -3
10y + y2 = 0 + -3

Combine like terms: 0 + -3 = -3
10y + y2 = -3

The y term is 10y.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10y + 25 + y2 = -3 + 25

Reorder the terms:
25 + 10y + y2 = -3 + 25

Combine like terms: -3 + 25 = 22
25 + 10y + y2 = 22

Factor a perfect square on the left side:
(y + 5)(y + 5) = 22

Calculate the square root of the right side: 4.69041576

Break this problem into two subproblems by setting 
(y + 5) equal to 4.69041576 and -4.69041576.

Subproblem 1

y + 5 = 4.69041576 Simplifying y + 5 = 4.69041576 Reorder the terms: 5 + y = 4.69041576 Solving 5 + y = 4.69041576 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + y = 4.69041576 + -5 Combine like terms: 5 + -5 = 0 0 + y = 4.69041576 + -5 y = 4.69041576 + -5 Combine like terms: 4.69041576 + -5 = -0.30958424 y = -0.30958424 Simplifying y = -0.30958424

Subproblem 2

y + 5 = -4.69041576 Simplifying y + 5 = -4.69041576 Reorder the terms: 5 + y = -4.69041576 Solving 5 + y = -4.69041576 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + y = -4.69041576 + -5 Combine like terms: 5 + -5 = 0 0 + y = -4.69041576 + -5 y = -4.69041576 + -5 Combine like terms: -4.69041576 + -5 = -9.69041576 y = -9.69041576 Simplifying y = -9.69041576

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-0.30958424, -9.69041576}

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