x^2-23x+75=0

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Solution for x^2-23x+75=0 equation:


Simplifying
x2 + -23x + 75 = 0

Reorder the terms:
75 + -23x + x2 = 0

Solving
75 + -23x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-75' to each side of the equation.
75 + -23x + -75 + x2 = 0 + -75

Reorder the terms:
75 + -75 + -23x + x2 = 0 + -75

Combine like terms: 75 + -75 = 0
0 + -23x + x2 = 0 + -75
-23x + x2 = 0 + -75

Combine like terms: 0 + -75 = -75
-23x + x2 = -75

The x term is -23x.  Take half its coefficient (-11.5).
Square it (132.25) and add it to both sides.

Add '132.25' to each side of the equation.
-23x + 132.25 + x2 = -75 + 132.25

Reorder the terms:
132.25 + -23x + x2 = -75 + 132.25

Combine like terms: -75 + 132.25 = 57.25
132.25 + -23x + x2 = 57.25

Factor a perfect square on the left side:
(x + -11.5)(x + -11.5) = 57.25

Calculate the square root of the right side: 7.566372975

Break this problem into two subproblems by setting 
(x + -11.5) equal to 7.566372975 and -7.566372975.

Subproblem 1

x + -11.5 = 7.566372975 Simplifying x + -11.5 = 7.566372975 Reorder the terms: -11.5 + x = 7.566372975 Solving -11.5 + x = 7.566372975 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + x = 7.566372975 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + x = 7.566372975 + 11.5 x = 7.566372975 + 11.5 Combine like terms: 7.566372975 + 11.5 = 19.066372975 x = 19.066372975 Simplifying x = 19.066372975

Subproblem 2

x + -11.5 = -7.566372975 Simplifying x + -11.5 = -7.566372975 Reorder the terms: -11.5 + x = -7.566372975 Solving -11.5 + x = -7.566372975 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + x = -7.566372975 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + x = -7.566372975 + 11.5 x = -7.566372975 + 11.5 Combine like terms: -7.566372975 + 11.5 = 3.933627025 x = 3.933627025 Simplifying x = 3.933627025

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.066372975, 3.933627025}

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