4x^2-61x+128=0

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Solution for 4x^2-61x+128=0 equation:


Simplifying
4x2 + -61x + 128 = 0

Reorder the terms:
128 + -61x + 4x2 = 0

Solving
128 + -61x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
32 + -15.25x + x2 = 0

Move the constant term to the right:

Add '-32' to each side of the equation.
32 + -15.25x + -32 + x2 = 0 + -32

Reorder the terms:
32 + -32 + -15.25x + x2 = 0 + -32

Combine like terms: 32 + -32 = 0
0 + -15.25x + x2 = 0 + -32
-15.25x + x2 = 0 + -32

Combine like terms: 0 + -32 = -32
-15.25x + x2 = -32

The x term is -15.25x.  Take half its coefficient (-7.625).
Square it (58.140625) and add it to both sides.

Add '58.140625' to each side of the equation.
-15.25x + 58.140625 + x2 = -32 + 58.140625

Reorder the terms:
58.140625 + -15.25x + x2 = -32 + 58.140625

Combine like terms: -32 + 58.140625 = 26.140625
58.140625 + -15.25x + x2 = 26.140625

Factor a perfect square on the left side:
(x + -7.625)(x + -7.625) = 26.140625

Calculate the square root of the right side: 5.112790334

Break this problem into two subproblems by setting 
(x + -7.625) equal to 5.112790334 and -5.112790334.

Subproblem 1

x + -7.625 = 5.112790334 Simplifying x + -7.625 = 5.112790334 Reorder the terms: -7.625 + x = 5.112790334 Solving -7.625 + x = 5.112790334 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.625' to each side of the equation. -7.625 + 7.625 + x = 5.112790334 + 7.625 Combine like terms: -7.625 + 7.625 = 0.000 0.000 + x = 5.112790334 + 7.625 x = 5.112790334 + 7.625 Combine like terms: 5.112790334 + 7.625 = 12.737790334 x = 12.737790334 Simplifying x = 12.737790334

Subproblem 2

x + -7.625 = -5.112790334 Simplifying x + -7.625 = -5.112790334 Reorder the terms: -7.625 + x = -5.112790334 Solving -7.625 + x = -5.112790334 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.625' to each side of the equation. -7.625 + 7.625 + x = -5.112790334 + 7.625 Combine like terms: -7.625 + 7.625 = 0.000 0.000 + x = -5.112790334 + 7.625 x = -5.112790334 + 7.625 Combine like terms: -5.112790334 + 7.625 = 2.512209666 x = 2.512209666 Simplifying x = 2.512209666

Solution

The solution to the problem is based on the solutions from the subproblems. x = {12.737790334, 2.512209666}

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