y^2+7y-19=0

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Solution for y^2+7y-19=0 equation:


Simplifying
y2 + 7y + -19 = 0

Reorder the terms:
-19 + 7y + y2 = 0

Solving
-19 + 7y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + 7y + 19 + y2 = 0 + 19

Reorder the terms:
-19 + 19 + 7y + y2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + 7y + y2 = 0 + 19
7y + y2 = 0 + 19

Combine like terms: 0 + 19 = 19
7y + y2 = 19

The y term is 7y.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7y + 12.25 + y2 = 19 + 12.25

Reorder the terms:
12.25 + 7y + y2 = 19 + 12.25

Combine like terms: 19 + 12.25 = 31.25
12.25 + 7y + y2 = 31.25

Factor a perfect square on the left side:
(y + 3.5)(y + 3.5) = 31.25

Calculate the square root of the right side: 5.590169944

Break this problem into two subproblems by setting 
(y + 3.5) equal to 5.590169944 and -5.590169944.

Subproblem 1

y + 3.5 = 5.590169944 Simplifying y + 3.5 = 5.590169944 Reorder the terms: 3.5 + y = 5.590169944 Solving 3.5 + y = 5.590169944 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + y = 5.590169944 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + y = 5.590169944 + -3.5 y = 5.590169944 + -3.5 Combine like terms: 5.590169944 + -3.5 = 2.090169944 y = 2.090169944 Simplifying y = 2.090169944

Subproblem 2

y + 3.5 = -5.590169944 Simplifying y + 3.5 = -5.590169944 Reorder the terms: 3.5 + y = -5.590169944 Solving 3.5 + y = -5.590169944 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + y = -5.590169944 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + y = -5.590169944 + -3.5 y = -5.590169944 + -3.5 Combine like terms: -5.590169944 + -3.5 = -9.090169944 y = -9.090169944 Simplifying y = -9.090169944

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.090169944, -9.090169944}

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