42-y^2-4y=9

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Solution for 42-y^2-4y=9 equation:


Simplifying
42 + -1y2 + -4y = 9

Reorder the terms:
42 + -4y + -1y2 = 9

Solving
42 + -4y + -1y2 = 9

Solving for variable 'y'.

Reorder the terms:
42 + -9 + -4y + -1y2 = 9 + -9

Combine like terms: 42 + -9 = 33
33 + -4y + -1y2 = 9 + -9

Combine like terms: 9 + -9 = 0
33 + -4y + -1y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-33 + 4y + y2 = 0

Move the constant term to the right:

Add '33' to each side of the equation.
-33 + 4y + 33 + y2 = 0 + 33

Reorder the terms:
-33 + 33 + 4y + y2 = 0 + 33

Combine like terms: -33 + 33 = 0
0 + 4y + y2 = 0 + 33
4y + y2 = 0 + 33

Combine like terms: 0 + 33 = 33
4y + y2 = 33

The y term is 4y.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4y + 4 + y2 = 33 + 4

Reorder the terms:
4 + 4y + y2 = 33 + 4

Combine like terms: 33 + 4 = 37
4 + 4y + y2 = 37

Factor a perfect square on the left side:
(y + 2)(y + 2) = 37

Calculate the square root of the right side: 6.08276253

Break this problem into two subproblems by setting 
(y + 2) equal to 6.08276253 and -6.08276253.

Subproblem 1

y + 2 = 6.08276253 Simplifying y + 2 = 6.08276253 Reorder the terms: 2 + y = 6.08276253 Solving 2 + y = 6.08276253 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + y = 6.08276253 + -2 Combine like terms: 2 + -2 = 0 0 + y = 6.08276253 + -2 y = 6.08276253 + -2 Combine like terms: 6.08276253 + -2 = 4.08276253 y = 4.08276253 Simplifying y = 4.08276253

Subproblem 2

y + 2 = -6.08276253 Simplifying y + 2 = -6.08276253 Reorder the terms: 2 + y = -6.08276253 Solving 2 + y = -6.08276253 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + y = -6.08276253 + -2 Combine like terms: 2 + -2 = 0 0 + y = -6.08276253 + -2 y = -6.08276253 + -2 Combine like terms: -6.08276253 + -2 = -8.08276253 y = -8.08276253 Simplifying y = -8.08276253

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.08276253, -8.08276253}

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