0=10x-15-x^2

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Solution for 0=10x-15-x^2 equation:


Simplifying
0 = 10x + -15 + -1x2

Reorder the terms:
0 = -15 + 10x + -1x2

Solving
0 = -15 + 10x + -1x2

Solving for variable 'x'.

Combine like terms: 0 + 15 = 15
15 + -10x + x2 = -15 + 10x + -1x2 + 15 + -10x + x2

Reorder the terms:
15 + -10x + x2 = -15 + 15 + 10x + -10x + -1x2 + x2

Combine like terms: -15 + 15 = 0
15 + -10x + x2 = 0 + 10x + -10x + -1x2 + x2
15 + -10x + x2 = 10x + -10x + -1x2 + x2

Combine like terms: 10x + -10x = 0
15 + -10x + x2 = 0 + -1x2 + x2
15 + -10x + x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
15 + -10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-15' to each side of the equation.
15 + -10x + -15 + x2 = 0 + -15

Reorder the terms:
15 + -15 + -10x + x2 = 0 + -15

Combine like terms: 15 + -15 = 0
0 + -10x + x2 = 0 + -15
-10x + x2 = 0 + -15

Combine like terms: 0 + -15 = -15
-10x + x2 = -15

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 = -15 + 25

Reorder the terms:
25 + -10x + x2 = -15 + 25

Combine like terms: -15 + 25 = 10
25 + -10x + x2 = 10

Factor a perfect square on the left side:
(x + -5)(x + -5) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(x + -5) equal to 3.16227766 and -3.16227766.

Subproblem 1

x + -5 = 3.16227766 Simplifying x + -5 = 3.16227766 Reorder the terms: -5 + x = 3.16227766 Solving -5 + x = 3.16227766 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 = 3.16227766 + 5 Combine like terms: -5 + 5 = 0 0 + x = 3.16227766 + 5 x = 3.16227766 + 5 Combine like terms: 3.16227766 + 5 = 8.16227766 x = 8.16227766 Simplifying x = 8.16227766

Subproblem 2

x + -5 = -3.16227766 Simplifying x + -5 = -3.16227766 Reorder the terms: -5 + x = -3.16227766 Solving -5 + x = -3.16227766 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 = -3.16227766 + 5 Combine like terms: -5 + 5 = 0 0 + x = -3.16227766 + 5 x = -3.16227766 + 5 Combine like terms: -3.16227766 + 5 = 1.83772234 x = 1.83772234 Simplifying x = 1.83772234

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.16227766, 1.83772234}

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