(2w-3)w=45

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Solution for (2w-3)w=45 equation:


Simplifying
(2w + -3) * w = 45

Reorder the terms:
(-3 + 2w) * w = 45

Reorder the terms for easier multiplication:
w(-3 + 2w) = 45
(-3 * w + 2w * w) = 45
(-3w + 2w2) = 45

Solving
-3w + 2w2 = 45

Solving for variable 'w'.

Reorder the terms:
-45 + -3w + 2w2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + -3w + 2w2 = 0

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

Divide each side by '2'.
-22.5 + -1.5w + w2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + -1.5w + 22.5 + w2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + -1.5w + w2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + -1.5w + w2 = 0 + 22.5
-1.5w + w2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
-1.5w + w2 = 22.5

The w term is -1.5w.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5w + 0.5625 + w2 = 22.5 + 0.5625

Reorder the terms:
0.5625 + -1.5w + w2 = 22.5 + 0.5625

Combine like terms: 22.5 + 0.5625 = 23.0625
0.5625 + -1.5w + w2 = 23.0625

Factor a perfect square on the left side:
(w + -0.75)(w + -0.75) = 23.0625

Calculate the square root of the right side: 4.802343178

Break this problem into two subproblems by setting 
(w + -0.75) equal to 4.802343178 and -4.802343178.

Subproblem 1

w + -0.75 = 4.802343178 Simplifying w + -0.75 = 4.802343178 Reorder the terms: -0.75 + w = 4.802343178 Solving -0.75 + w = 4.802343178 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + w = 4.802343178 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + w = 4.802343178 + 0.75 w = 4.802343178 + 0.75 Combine like terms: 4.802343178 + 0.75 = 5.552343178 w = 5.552343178 Simplifying w = 5.552343178

Subproblem 2

w + -0.75 = -4.802343178 Simplifying w + -0.75 = -4.802343178 Reorder the terms: -0.75 + w = -4.802343178 Solving -0.75 + w = -4.802343178 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + w = -4.802343178 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + w = -4.802343178 + 0.75 w = -4.802343178 + 0.75 Combine like terms: -4.802343178 + 0.75 = -4.052343178 w = -4.052343178 Simplifying w = -4.052343178

Solution

The solution to the problem is based on the solutions from the subproblems. w = {5.552343178, -4.052343178}

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