24-w*w=3w-8

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Solution for 24-w*w=3w-8 equation:


Simplifying
24 + -1w * w = 3w + -8

Multiply w * w
24 + -1w2 = 3w + -8

Reorder the terms:
24 + -1w2 = -8 + 3w

Solving
24 + -1w2 = -8 + 3w

Solving for variable 'w'.

Reorder the terms:
24 + 8 + -3w + -1w2 = -8 + 3w + 8 + -3w

Combine like terms: 24 + 8 = 32
32 + -3w + -1w2 = -8 + 3w + 8 + -3w

Reorder the terms:
32 + -3w + -1w2 = -8 + 8 + 3w + -3w

Combine like terms: -8 + 8 = 0
32 + -3w + -1w2 = 0 + 3w + -3w
32 + -3w + -1w2 = 3w + -3w

Combine like terms: 3w + -3w = 0
32 + -3w + -1w2 = 0

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

Divide each side by '-1'.
-32 + 3w + w2 = 0

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + 3w + 32 + w2 = 0 + 32

Reorder the terms:
-32 + 32 + 3w + w2 = 0 + 32

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

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

The w term is 3w.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3w + 2.25 + w2 = 32 + 2.25

Reorder the terms:
2.25 + 3w + w2 = 32 + 2.25

Combine like terms: 32 + 2.25 = 34.25
2.25 + 3w + w2 = 34.25

Factor a perfect square on the left side:
(w + 1.5)(w + 1.5) = 34.25

Calculate the square root of the right side: 5.852349955

Break this problem into two subproblems by setting 
(w + 1.5) equal to 5.852349955 and -5.852349955.

Subproblem 1

w + 1.5 = 5.852349955 Simplifying w + 1.5 = 5.852349955 Reorder the terms: 1.5 + w = 5.852349955 Solving 1.5 + w = 5.852349955 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 5.852349955 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 5.852349955 + -1.5 w = 5.852349955 + -1.5 Combine like terms: 5.852349955 + -1.5 = 4.352349955 w = 4.352349955 Simplifying w = 4.352349955

Subproblem 2

w + 1.5 = -5.852349955 Simplifying w + 1.5 = -5.852349955 Reorder the terms: 1.5 + w = -5.852349955 Solving 1.5 + w = -5.852349955 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -5.852349955 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -5.852349955 + -1.5 w = -5.852349955 + -1.5 Combine like terms: -5.852349955 + -1.5 = -7.352349955 w = -7.352349955 Simplifying w = -7.352349955

Solution

The solution to the problem is based on the solutions from the subproblems. w = {4.352349955, -7.352349955}

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