21y^2+10y-19=0

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Solution for 21y^2+10y-19=0 equation:


Simplifying
21y2 + 10y + -19 = 0

Reorder the terms:
-19 + 10y + 21y2 = 0

Solving
-19 + 10y + 21y2 = 0

Solving for variable 'y'.

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

Divide each side by '21'.
-0.9047619048 + 0.4761904762y + y2 = 0

Move the constant term to the right:

Add '0.9047619048' to each side of the equation.
-0.9047619048 + 0.4761904762y + 0.9047619048 + y2 = 0 + 0.9047619048

Reorder the terms:
-0.9047619048 + 0.9047619048 + 0.4761904762y + y2 = 0 + 0.9047619048

Combine like terms: -0.9047619048 + 0.9047619048 = 0.0000000000
0.0000000000 + 0.4761904762y + y2 = 0 + 0.9047619048
0.4761904762y + y2 = 0 + 0.9047619048

Combine like terms: 0 + 0.9047619048 = 0.9047619048
0.4761904762y + y2 = 0.9047619048

The y term is 0.4761904762y.  Take half its coefficient (0.2380952381).
Square it (0.05668934241) and add it to both sides.

Add '0.05668934241' to each side of the equation.
0.4761904762y + 0.05668934241 + y2 = 0.9047619048 + 0.05668934241

Reorder the terms:
0.05668934241 + 0.4761904762y + y2 = 0.9047619048 + 0.05668934241

Combine like terms: 0.9047619048 + 0.05668934241 = 0.96145124721
0.05668934241 + 0.4761904762y + y2 = 0.96145124721

Factor a perfect square on the left side:
(y + 0.2380952381)(y + 0.2380952381) = 0.96145124721

Calculate the square root of the right side: 0.980536204

Break this problem into two subproblems by setting 
(y + 0.2380952381) equal to 0.980536204 and -0.980536204.

Subproblem 1

y + 0.2380952381 = 0.980536204 Simplifying y + 0.2380952381 = 0.980536204 Reorder the terms: 0.2380952381 + y = 0.980536204 Solving 0.2380952381 + y = 0.980536204 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.2380952381' to each side of the equation. 0.2380952381 + -0.2380952381 + y = 0.980536204 + -0.2380952381 Combine like terms: 0.2380952381 + -0.2380952381 = 0.0000000000 0.0000000000 + y = 0.980536204 + -0.2380952381 y = 0.980536204 + -0.2380952381 Combine like terms: 0.980536204 + -0.2380952381 = 0.7424409659 y = 0.7424409659 Simplifying y = 0.7424409659

Subproblem 2

y + 0.2380952381 = -0.980536204 Simplifying y + 0.2380952381 = -0.980536204 Reorder the terms: 0.2380952381 + y = -0.980536204 Solving 0.2380952381 + y = -0.980536204 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.2380952381' to each side of the equation. 0.2380952381 + -0.2380952381 + y = -0.980536204 + -0.2380952381 Combine like terms: 0.2380952381 + -0.2380952381 = 0.0000000000 0.0000000000 + y = -0.980536204 + -0.2380952381 y = -0.980536204 + -0.2380952381 Combine like terms: -0.980536204 + -0.2380952381 = -1.2186314421 y = -1.2186314421 Simplifying y = -1.2186314421

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.7424409659, -1.2186314421}

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