5y^2-4y-21=0

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Solution for 5y^2-4y-21=0 equation:


Simplifying
5y2 + -4y + -21 = 0

Reorder the terms:
-21 + -4y + 5y2 = 0

Solving
-21 + -4y + 5y2 = 0

Solving for variable 'y'.

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

Divide each side by '5'.
-4.2 + -0.8y + y2 = 0

Move the constant term to the right:

Add '4.2' to each side of the equation.
-4.2 + -0.8y + 4.2 + y2 = 0 + 4.2

Reorder the terms:
-4.2 + 4.2 + -0.8y + y2 = 0 + 4.2

Combine like terms: -4.2 + 4.2 = 0.0
0.0 + -0.8y + y2 = 0 + 4.2
-0.8y + y2 = 0 + 4.2

Combine like terms: 0 + 4.2 = 4.2
-0.8y + y2 = 4.2

The y term is -0.8y.  Take half its coefficient (-0.4).
Square it (0.16) and add it to both sides.

Add '0.16' to each side of the equation.
-0.8y + 0.16 + y2 = 4.2 + 0.16

Reorder the terms:
0.16 + -0.8y + y2 = 4.2 + 0.16

Combine like terms: 4.2 + 0.16 = 4.36
0.16 + -0.8y + y2 = 4.36

Factor a perfect square on the left side:
(y + -0.4)(y + -0.4) = 4.36

Calculate the square root of the right side: 2.088061302

Break this problem into two subproblems by setting 
(y + -0.4) equal to 2.088061302 and -2.088061302.

Subproblem 1

y + -0.4 = 2.088061302 Simplifying y + -0.4 = 2.088061302 Reorder the terms: -0.4 + y = 2.088061302 Solving -0.4 + y = 2.088061302 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + y = 2.088061302 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + y = 2.088061302 + 0.4 y = 2.088061302 + 0.4 Combine like terms: 2.088061302 + 0.4 = 2.488061302 y = 2.488061302 Simplifying y = 2.488061302

Subproblem 2

y + -0.4 = -2.088061302 Simplifying y + -0.4 = -2.088061302 Reorder the terms: -0.4 + y = -2.088061302 Solving -0.4 + y = -2.088061302 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + y = -2.088061302 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + y = -2.088061302 + 0.4 y = -2.088061302 + 0.4 Combine like terms: -2.088061302 + 0.4 = -1.688061302 y = -1.688061302 Simplifying y = -1.688061302

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.488061302, -1.688061302}

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