(4x^2)-(52x)+161=0

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Solution for (4x^2)-(52x)+161=0 equation:


Simplifying
(4x2) + -1(52x) + 161 = 0

Remove parenthesis around (52x)
(4x2) + -1 * 52x + 161 = 0

Multiply -1 * 52
(4x2) + -52x + 161 = 0

Reorder the terms:
161 + -52x + (4x2) = 0

Solving
161 + -52x + (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'.
40.25 + -13x + x2 = 0

Move the constant term to the right:

Add '-40.25' to each side of the equation.
40.25 + -13x + -40.25 + x2 = 0 + -40.25

Reorder the terms:
40.25 + -40.25 + -13x + x2 = 0 + -40.25

Combine like terms: 40.25 + -40.25 = 0.00
0.00 + -13x + x2 = 0 + -40.25
-13x + x2 = 0 + -40.25

Combine like terms: 0 + -40.25 = -40.25
-13x + x2 = -40.25

The x term is -13x.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13x + 42.25 + x2 = -40.25 + 42.25

Reorder the terms:
42.25 + -13x + x2 = -40.25 + 42.25

Combine like terms: -40.25 + 42.25 = 2
42.25 + -13x + x2 = 2

Factor a perfect square on the left side:
((x) + -6.5)((x) + -6.5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
((x) + -6.5) equal to 1.414213562 and -1.414213562.

Subproblem 1

(x) + -6.5 = 1.414213562 Simplifying (x) + -6.5 = 1.414213562 x + -6.5 = 1.414213562 Reorder the terms: -6.5 + x = 1.414213562 Solving -6.5 + x = 1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x = 1.414213562 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x = 1.414213562 + 6.5 x = 1.414213562 + 6.5 Combine like terms: 1.414213562 + 6.5 = 7.914213562 x = 7.914213562 Simplifying x = 7.914213562

Subproblem 2

(x) + -6.5 = -1.414213562 Simplifying (x) + -6.5 = -1.414213562 x + -6.5 = -1.414213562 Reorder the terms: -6.5 + x = -1.414213562 Solving -6.5 + x = -1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x = -1.414213562 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x = -1.414213562 + 6.5 x = -1.414213562 + 6.5 Combine like terms: -1.414213562 + 6.5 = 5.085786438 x = 5.085786438 Simplifying x = 5.085786438

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.914213562, 5.085786438}

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