10x+1=289-x^2

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Solution for 10x+1=289-x^2 equation:


Simplifying
10x + 1 = 289 + -1x2

Reorder the terms:
1 + 10x = 289 + -1x2

Solving
1 + 10x = 289 + -1x2

Solving for variable 'x'.

Reorder the terms:
1 + -289 + 10x + x2 = 289 + -1x2 + -289 + x2

Combine like terms: 1 + -289 = -288
-288 + 10x + x2 = 289 + -1x2 + -289 + x2

Reorder the terms:
-288 + 10x + x2 = 289 + -289 + -1x2 + x2

Combine like terms: 289 + -289 = 0
-288 + 10x + x2 = 0 + -1x2 + x2
-288 + 10x + x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
-288 + 10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '288' to each side of the equation.
-288 + 10x + 288 + x2 = 0 + 288

Reorder the terms:
-288 + 288 + 10x + x2 = 0 + 288

Combine like terms: -288 + 288 = 0
0 + 10x + x2 = 0 + 288
10x + x2 = 0 + 288

Combine like terms: 0 + 288 = 288
10x + x2 = 288

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 = 288 + 25

Reorder the terms:
25 + 10x + x2 = 288 + 25

Combine like terms: 288 + 25 = 313
25 + 10x + x2 = 313

Factor a perfect square on the left side:
(x + 5)(x + 5) = 313

Calculate the square root of the right side: 17.691806013

Break this problem into two subproblems by setting 
(x + 5) equal to 17.691806013 and -17.691806013.

Subproblem 1

x + 5 = 17.691806013 Simplifying x + 5 = 17.691806013 Reorder the terms: 5 + x = 17.691806013 Solving 5 + x = 17.691806013 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 = 17.691806013 + -5 Combine like terms: 5 + -5 = 0 0 + x = 17.691806013 + -5 x = 17.691806013 + -5 Combine like terms: 17.691806013 + -5 = 12.691806013 x = 12.691806013 Simplifying x = 12.691806013

Subproblem 2

x + 5 = -17.691806013 Simplifying x + 5 = -17.691806013 Reorder the terms: 5 + x = -17.691806013 Solving 5 + x = -17.691806013 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 = -17.691806013 + -5 Combine like terms: 5 + -5 = 0 0 + x = -17.691806013 + -5 x = -17.691806013 + -5 Combine like terms: -17.691806013 + -5 = -22.691806013 x = -22.691806013 Simplifying x = -22.691806013

Solution

The solution to the problem is based on the solutions from the subproblems. x = {12.691806013, -22.691806013}

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