4w^2-4w+1=75

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Solution for 4w^2-4w+1=75 equation:


Simplifying
4w2 + -4w + 1 = 75

Reorder the terms:
1 + -4w + 4w2 = 75

Solving
1 + -4w + 4w2 = 75

Solving for variable 'w'.

Reorder the terms:
1 + -75 + -4w + 4w2 = 75 + -75

Combine like terms: 1 + -75 = -74
-74 + -4w + 4w2 = 75 + -75

Combine like terms: 75 + -75 = 0
-74 + -4w + 4w2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-37 + -2w + 2w2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(-37 + -2w + 2w2)' equal to zero and attempt to solve: Simplifying -37 + -2w + 2w2 = 0 Solving -37 + -2w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -18.5 + -1w + w2 = 0 Move the constant term to the right: Add '18.5' to each side of the equation. -18.5 + -1w + 18.5 + w2 = 0 + 18.5 Reorder the terms: -18.5 + 18.5 + -1w + w2 = 0 + 18.5 Combine like terms: -18.5 + 18.5 = 0.0 0.0 + -1w + w2 = 0 + 18.5 -1w + w2 = 0 + 18.5 Combine like terms: 0 + 18.5 = 18.5 -1w + w2 = 18.5 The w term is -1w. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1w + 0.25 + w2 = 18.5 + 0.25 Reorder the terms: 0.25 + -1w + w2 = 18.5 + 0.25 Combine like terms: 18.5 + 0.25 = 18.75 0.25 + -1w + w2 = 18.75 Factor a perfect square on the left side: (w + -0.5)(w + -0.5) = 18.75 Calculate the square root of the right side: 4.330127019 Break this problem into two subproblems by setting (w + -0.5) equal to 4.330127019 and -4.330127019.

Subproblem 1

w + -0.5 = 4.330127019 Simplifying w + -0.5 = 4.330127019 Reorder the terms: -0.5 + w = 4.330127019 Solving -0.5 + w = 4.330127019 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + w = 4.330127019 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + w = 4.330127019 + 0.5 w = 4.330127019 + 0.5 Combine like terms: 4.330127019 + 0.5 = 4.830127019 w = 4.830127019 Simplifying w = 4.830127019

Subproblem 2

w + -0.5 = -4.330127019 Simplifying w + -0.5 = -4.330127019 Reorder the terms: -0.5 + w = -4.330127019 Solving -0.5 + w = -4.330127019 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + w = -4.330127019 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + w = -4.330127019 + 0.5 w = -4.330127019 + 0.5 Combine like terms: -4.330127019 + 0.5 = -3.830127019 w = -3.830127019 Simplifying w = -3.830127019

Solution

The solution to the problem is based on the solutions from the subproblems. w = {4.830127019, -3.830127019}

Solution

w = {4.830127019, -3.830127019}

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