13x+x^2=75

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Solution for 13x+x^2=75 equation:


Simplifying
13x + x2 = 75

Solving
13x + x2 = 75

Solving for variable 'x'.

Reorder the terms:
-75 + 13x + x2 = 75 + -75

Combine like terms: 75 + -75 = 0
-75 + 13x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '75' to each side of the equation.
-75 + 13x + 75 + x2 = 0 + 75

Reorder the terms:
-75 + 75 + 13x + x2 = 0 + 75

Combine like terms: -75 + 75 = 0
0 + 13x + x2 = 0 + 75
13x + x2 = 0 + 75

Combine like terms: 0 + 75 = 75
13x + x2 = 75

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 = 75 + 42.25

Reorder the terms:
42.25 + 13x + x2 = 75 + 42.25

Combine like terms: 75 + 42.25 = 117.25
42.25 + 13x + x2 = 117.25

Factor a perfect square on the left side:
(x + 6.5)(x + 6.5) = 117.25

Calculate the square root of the right side: 10.828203914

Break this problem into two subproblems by setting 
(x + 6.5) equal to 10.828203914 and -10.828203914.

Subproblem 1

x + 6.5 = 10.828203914 Simplifying x + 6.5 = 10.828203914 Reorder the terms: 6.5 + x = 10.828203914 Solving 6.5 + x = 10.828203914 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 = 10.828203914 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = 10.828203914 + -6.5 x = 10.828203914 + -6.5 Combine like terms: 10.828203914 + -6.5 = 4.328203914 x = 4.328203914 Simplifying x = 4.328203914

Subproblem 2

x + 6.5 = -10.828203914 Simplifying x + 6.5 = -10.828203914 Reorder the terms: 6.5 + x = -10.828203914 Solving 6.5 + x = -10.828203914 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 = -10.828203914 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = -10.828203914 + -6.5 x = -10.828203914 + -6.5 Combine like terms: -10.828203914 + -6.5 = -17.328203914 x = -17.328203914 Simplifying x = -17.328203914

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.328203914, -17.328203914}

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