4x^2-42x+13=0

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Solution for 4x^2-42x+13=0 equation:


Simplifying
4x2 + -42x + 13 = 0

Reorder the terms:
13 + -42x + 4x2 = 0

Solving
13 + -42x + 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'.
3.25 + -10.5x + x2 = 0

Move the constant term to the right:

Add '-3.25' to each side of the equation.
3.25 + -10.5x + -3.25 + x2 = 0 + -3.25

Reorder the terms:
3.25 + -3.25 + -10.5x + x2 = 0 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + -10.5x + x2 = 0 + -3.25
-10.5x + x2 = 0 + -3.25

Combine like terms: 0 + -3.25 = -3.25
-10.5x + x2 = -3.25

The x term is -10.5x.  Take half its coefficient (-5.25).
Square it (27.5625) and add it to both sides.

Add '27.5625' to each side of the equation.
-10.5x + 27.5625 + x2 = -3.25 + 27.5625

Reorder the terms:
27.5625 + -10.5x + x2 = -3.25 + 27.5625

Combine like terms: -3.25 + 27.5625 = 24.3125
27.5625 + -10.5x + x2 = 24.3125

Factor a perfect square on the left side:
(x + -5.25)(x + -5.25) = 24.3125

Calculate the square root of the right side: 4.930770731

Break this problem into two subproblems by setting 
(x + -5.25) equal to 4.930770731 and -4.930770731.

Subproblem 1

x + -5.25 = 4.930770731 Simplifying x + -5.25 = 4.930770731 Reorder the terms: -5.25 + x = 4.930770731 Solving -5.25 + x = 4.930770731 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.25' to each side of the equation. -5.25 + 5.25 + x = 4.930770731 + 5.25 Combine like terms: -5.25 + 5.25 = 0.00 0.00 + x = 4.930770731 + 5.25 x = 4.930770731 + 5.25 Combine like terms: 4.930770731 + 5.25 = 10.180770731 x = 10.180770731 Simplifying x = 10.180770731

Subproblem 2

x + -5.25 = -4.930770731 Simplifying x + -5.25 = -4.930770731 Reorder the terms: -5.25 + x = -4.930770731 Solving -5.25 + x = -4.930770731 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.25' to each side of the equation. -5.25 + 5.25 + x = -4.930770731 + 5.25 Combine like terms: -5.25 + 5.25 = 0.00 0.00 + x = -4.930770731 + 5.25 x = -4.930770731 + 5.25 Combine like terms: -4.930770731 + 5.25 = 0.319229269 x = 0.319229269 Simplifying x = 0.319229269

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10.180770731, 0.319229269}

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