4x^2-32x+61=0

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Solution for 4x^2-32x+61=0 equation:


Simplifying
4x2 + -32x + 61 = 0

Reorder the terms:
61 + -32x + 4x2 = 0

Solving
61 + -32x + 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'.
15.25 + -8x + x2 = 0

Move the constant term to the right:

Add '-15.25' to each side of the equation.
15.25 + -8x + -15.25 + x2 = 0 + -15.25

Reorder the terms:
15.25 + -15.25 + -8x + x2 = 0 + -15.25

Combine like terms: 15.25 + -15.25 = 0.00
0.00 + -8x + x2 = 0 + -15.25
-8x + x2 = 0 + -15.25

Combine like terms: 0 + -15.25 = -15.25
-8x + x2 = -15.25

The x term is -8x.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x + 16 + x2 = -15.25 + 16

Reorder the terms:
16 + -8x + x2 = -15.25 + 16

Combine like terms: -15.25 + 16 = 0.75
16 + -8x + x2 = 0.75

Factor a perfect square on the left side:
(x + -4)(x + -4) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(x + -4) equal to 0.866025404 and -0.866025404.

Subproblem 1

x + -4 = 0.866025404 Simplifying x + -4 = 0.866025404 Reorder the terms: -4 + x = 0.866025404 Solving -4 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = 0.866025404 + 4 Combine like terms: -4 + 4 = 0 0 + x = 0.866025404 + 4 x = 0.866025404 + 4 Combine like terms: 0.866025404 + 4 = 4.866025404 x = 4.866025404 Simplifying x = 4.866025404

Subproblem 2

x + -4 = -0.866025404 Simplifying x + -4 = -0.866025404 Reorder the terms: -4 + x = -0.866025404 Solving -4 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = -0.866025404 + 4 Combine like terms: -4 + 4 = 0 0 + x = -0.866025404 + 4 x = -0.866025404 + 4 Combine like terms: -0.866025404 + 4 = 3.133974596 x = 3.133974596 Simplifying x = 3.133974596

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.866025404, 3.133974596}

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