4y^2-8y+2=y^2+7

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Solution for 4y^2-8y+2=y^2+7 equation:


Simplifying
4y2 + -8y + 2 = y2 + 7

Reorder the terms:
2 + -8y + 4y2 = y2 + 7

Reorder the terms:
2 + -8y + 4y2 = 7 + y2

Solving
2 + -8y + 4y2 = 7 + y2

Solving for variable 'y'.

Reorder the terms:
2 + -7 + -8y + 4y2 + -1y2 = 7 + y2 + -7 + -1y2

Combine like terms: 2 + -7 = -5
-5 + -8y + 4y2 + -1y2 = 7 + y2 + -7 + -1y2

Combine like terms: 4y2 + -1y2 = 3y2
-5 + -8y + 3y2 = 7 + y2 + -7 + -1y2

Reorder the terms:
-5 + -8y + 3y2 = 7 + -7 + y2 + -1y2

Combine like terms: 7 + -7 = 0
-5 + -8y + 3y2 = 0 + y2 + -1y2
-5 + -8y + 3y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
-5 + -8y + 3y2 = 0

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

Divide each side by '3'.
-1.666666667 + -2.666666667y + y2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -2.666666667y + 1.666666667 + y2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -2.666666667y + y2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -2.666666667y + y2 = 0 + 1.666666667
-2.666666667y + y2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-2.666666667y + y2 = 1.666666667

The y term is -2.666666667y.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667y + 1.777777780 + y2 = 1.666666667 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667y + y2 = 1.666666667 + 1.777777780

Combine like terms: 1.666666667 + 1.777777780 = 3.444444447
1.777777780 + -2.666666667y + y2 = 3.444444447

Factor a perfect square on the left side:
(y + -1.333333334)(y + -1.333333334) = 3.444444447

Calculate the square root of the right side: 1.855921455

Break this problem into two subproblems by setting 
(y + -1.333333334) equal to 1.855921455 and -1.855921455.

Subproblem 1

y + -1.333333334 = 1.855921455 Simplifying y + -1.333333334 = 1.855921455 Reorder the terms: -1.333333334 + y = 1.855921455 Solving -1.333333334 + y = 1.855921455 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = 1.855921455 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = 1.855921455 + 1.333333334 y = 1.855921455 + 1.333333334 Combine like terms: 1.855921455 + 1.333333334 = 3.189254789 y = 3.189254789 Simplifying y = 3.189254789

Subproblem 2

y + -1.333333334 = -1.855921455 Simplifying y + -1.333333334 = -1.855921455 Reorder the terms: -1.333333334 + y = -1.855921455 Solving -1.333333334 + y = -1.855921455 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = -1.855921455 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = -1.855921455 + 1.333333334 y = -1.855921455 + 1.333333334 Combine like terms: -1.855921455 + 1.333333334 = -0.522588121 y = -0.522588121 Simplifying y = -0.522588121

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.189254789, -0.522588121}

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