10x-x^2=13

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Solution for 10x-x^2=13 equation:


Simplifying
10x + -1x2 = 13

Solving
10x + -1x2 = 13

Solving for variable 'x'.

Reorder the terms:
-13 + 10x + -1x2 = 13 + -13

Combine like terms: 13 + -13 = 0
-13 + 10x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
13 + -10x + x2 = 0

Move the constant term to the right:

Add '-13' to each side of the equation.
13 + -10x + -13 + x2 = 0 + -13

Reorder the terms:
13 + -13 + -10x + x2 = 0 + -13

Combine like terms: 13 + -13 = 0
0 + -10x + x2 = 0 + -13
-10x + x2 = 0 + -13

Combine like terms: 0 + -13 = -13
-10x + x2 = -13

The x term is -10x.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x + 25 + x2 = -13 + 25

Reorder the terms:
25 + -10x + x2 = -13 + 25

Combine like terms: -13 + 25 = 12
25 + -10x + x2 = 12

Factor a perfect square on the left side:
(x + -5)(x + -5) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(x + -5) equal to 3.464101615 and -3.464101615.

Subproblem 1

x + -5 = 3.464101615 Simplifying x + -5 = 3.464101615 Reorder the terms: -5 + x = 3.464101615 Solving -5 + x = 3.464101615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 3.464101615 + 5 Combine like terms: -5 + 5 = 0 0 + x = 3.464101615 + 5 x = 3.464101615 + 5 Combine like terms: 3.464101615 + 5 = 8.464101615 x = 8.464101615 Simplifying x = 8.464101615

Subproblem 2

x + -5 = -3.464101615 Simplifying x + -5 = -3.464101615 Reorder the terms: -5 + x = -3.464101615 Solving -5 + x = -3.464101615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -3.464101615 + 5 Combine like terms: -5 + 5 = 0 0 + x = -3.464101615 + 5 x = -3.464101615 + 5 Combine like terms: -3.464101615 + 5 = 1.535898385 x = 1.535898385 Simplifying x = 1.535898385

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.464101615, 1.535898385}

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