4y+3y^2=21

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Solution for 4y+3y^2=21 equation:


Simplifying
4y + 3y2 = 21

Solving
4y + 3y2 = 21

Solving for variable 'y'.

Reorder the terms:
-21 + 4y + 3y2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 4y + 3y2 = 0

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

Divide each side by '3'.
-7 + 1.333333333y + y2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 1.333333333y + 7 + y2 = 0 + 7

Reorder the terms:
-7 + 7 + 1.333333333y + y2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 1.333333333y + y2 = 0 + 7
1.333333333y + y2 = 0 + 7

Combine like terms: 0 + 7 = 7
1.333333333y + y2 = 7

The y term is 1.333333333y.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333y + 0.4444444442 + y2 = 7 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333y + y2 = 7 + 0.4444444442

Combine like terms: 7 + 0.4444444442 = 7.4444444442
0.4444444442 + 1.333333333y + y2 = 7.4444444442

Factor a perfect square on the left side:
(y + 0.6666666665)(y + 0.6666666665) = 7.4444444442

Calculate the square root of the right side: 2.728450924

Break this problem into two subproblems by setting 
(y + 0.6666666665) equal to 2.728450924 and -2.728450924.

Subproblem 1

y + 0.6666666665 = 2.728450924 Simplifying y + 0.6666666665 = 2.728450924 Reorder the terms: 0.6666666665 + y = 2.728450924 Solving 0.6666666665 + y = 2.728450924 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + y = 2.728450924 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + y = 2.728450924 + -0.6666666665 y = 2.728450924 + -0.6666666665 Combine like terms: 2.728450924 + -0.6666666665 = 2.0617842575 y = 2.0617842575 Simplifying y = 2.0617842575

Subproblem 2

y + 0.6666666665 = -2.728450924 Simplifying y + 0.6666666665 = -2.728450924 Reorder the terms: 0.6666666665 + y = -2.728450924 Solving 0.6666666665 + y = -2.728450924 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + y = -2.728450924 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + y = -2.728450924 + -0.6666666665 y = -2.728450924 + -0.6666666665 Combine like terms: -2.728450924 + -0.6666666665 = -3.3951175905 y = -3.3951175905 Simplifying y = -3.3951175905

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.0617842575, -3.3951175905}

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