4y^2-6y=-1

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Solution for 4y^2-6y=-1 equation:


Simplifying
4y2 + -6y = -1

Reorder the terms:
-6y + 4y2 = -1

Solving
-6y + 4y2 = -1

Solving for variable 'y'.

Reorder the terms:
1 + -6y + 4y2 = -1 + 1

Combine like terms: -1 + 1 = 0
1 + -6y + 4y2 = 0

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

Divide each side by '4'.
0.25 + -1.5y + y2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -1.5y + -0.25 + y2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -1.5y + y2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -1.5y + y2 = 0 + -0.25
-1.5y + y2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-1.5y + y2 = -0.25

The y term is -1.5y.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5y + 0.5625 + y2 = -0.25 + 0.5625

Reorder the terms:
0.5625 + -1.5y + y2 = -0.25 + 0.5625

Combine like terms: -0.25 + 0.5625 = 0.3125
0.5625 + -1.5y + y2 = 0.3125

Factor a perfect square on the left side:
(y + -0.75)(y + -0.75) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(y + -0.75) equal to 0.559016994 and -0.559016994.

Subproblem 1

y + -0.75 = 0.559016994 Simplifying y + -0.75 = 0.559016994 Reorder the terms: -0.75 + y = 0.559016994 Solving -0.75 + y = 0.559016994 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = 0.559016994 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = 0.559016994 + 0.75 y = 0.559016994 + 0.75 Combine like terms: 0.559016994 + 0.75 = 1.309016994 y = 1.309016994 Simplifying y = 1.309016994

Subproblem 2

y + -0.75 = -0.559016994 Simplifying y + -0.75 = -0.559016994 Reorder the terms: -0.75 + y = -0.559016994 Solving -0.75 + y = -0.559016994 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = -0.559016994 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = -0.559016994 + 0.75 y = -0.559016994 + 0.75 Combine like terms: -0.559016994 + 0.75 = 0.190983006 y = 0.190983006 Simplifying y = 0.190983006

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.309016994, 0.190983006}

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