3w^2-8w=32

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Solution for 3w^2-8w=32 equation:


Simplifying
3w2 + -8w = 32

Reorder the terms:
-8w + 3w2 = 32

Solving
-8w + 3w2 = 32

Solving for variable 'w'.

Reorder the terms:
-32 + -8w + 3w2 = 32 + -32

Combine like terms: 32 + -32 = 0
-32 + -8w + 3w2 = 0

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

Divide each side by '3'.
-10.66666667 + -2.666666667w + w2 = 0

Move the constant term to the right:

Add '10.66666667' to each side of the equation.
-10.66666667 + -2.666666667w + 10.66666667 + w2 = 0 + 10.66666667

Reorder the terms:
-10.66666667 + 10.66666667 + -2.666666667w + w2 = 0 + 10.66666667

Combine like terms: -10.66666667 + 10.66666667 = 0.00000000
0.00000000 + -2.666666667w + w2 = 0 + 10.66666667
-2.666666667w + w2 = 0 + 10.66666667

Combine like terms: 0 + 10.66666667 = 10.66666667
-2.666666667w + w2 = 10.66666667

The w term is -2.666666667w.  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.
-2.666666667w + 1.777777780 + w2 = 10.66666667 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667w + w2 = 10.66666667 + 1.777777780

Combine like terms: 10.66666667 + 1.777777780 = 12.44444445
1.777777780 + -2.666666667w + w2 = 12.44444445

Factor a perfect square on the left side:
(w + -1.333333334)(w + -1.333333334) = 12.44444445

Calculate the square root of the right side: 3.527668416

Break this problem into two subproblems by setting 
(w + -1.333333334) equal to 3.527668416 and -3.527668416.

Subproblem 1

w + -1.333333334 = 3.527668416 Simplifying w + -1.333333334 = 3.527668416 Reorder the terms: -1.333333334 + w = 3.527668416 Solving -1.333333334 + w = 3.527668416 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + w = 3.527668416 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + w = 3.527668416 + 1.333333334 w = 3.527668416 + 1.333333334 Combine like terms: 3.527668416 + 1.333333334 = 4.86100175 w = 4.86100175 Simplifying w = 4.86100175

Subproblem 2

w + -1.333333334 = -3.527668416 Simplifying w + -1.333333334 = -3.527668416 Reorder the terms: -1.333333334 + w = -3.527668416 Solving -1.333333334 + w = -3.527668416 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + w = -3.527668416 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + w = -3.527668416 + 1.333333334 w = -3.527668416 + 1.333333334 Combine like terms: -3.527668416 + 1.333333334 = -2.194335082 w = -2.194335082 Simplifying w = -2.194335082

Solution

The solution to the problem is based on the solutions from the subproblems. w = {4.86100175, -2.194335082}

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