7x=x^2+12-3x

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Solution for 7x=x^2+12-3x equation:


Simplifying
7x = x2 + 12 + -3x

Reorder the terms:
7x = 12 + -3x + x2

Solving
7x = 12 + -3x + x2

Solving for variable 'x'.

Reorder the terms:
-12 + 7x + 3x + -1x2 = 12 + -3x + x2 + -12 + 3x + -1x2

Combine like terms: 7x + 3x = 10x
-12 + 10x + -1x2 = 12 + -3x + x2 + -12 + 3x + -1x2

Reorder the terms:
-12 + 10x + -1x2 = 12 + -12 + -3x + 3x + x2 + -1x2

Combine like terms: 12 + -12 = 0
-12 + 10x + -1x2 = 0 + -3x + 3x + x2 + -1x2
-12 + 10x + -1x2 = -3x + 3x + x2 + -1x2

Combine like terms: -3x + 3x = 0
-12 + 10x + -1x2 = 0 + x2 + -1x2
-12 + 10x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-12 + 10x + -1x2 = 0

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

Divide each side by '-1'.
12 + -10x + x2 = 0

Move the constant term to the right:

Add '-12' to each side of the equation.
12 + -10x + -12 + x2 = 0 + -12

Reorder the terms:
12 + -12 + -10x + x2 = 0 + -12

Combine like terms: 12 + -12 = 0
0 + -10x + x2 = 0 + -12
-10x + x2 = 0 + -12

Combine like terms: 0 + -12 = -12
-10x + x2 = -12

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 = -12 + 25

Reorder the terms:
25 + -10x + x2 = -12 + 25

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

Factor a perfect square on the left side:
(x + -5)(x + -5) = 13

Calculate the square root of the right side: 3.605551275

Break this problem into two subproblems by setting 
(x + -5) equal to 3.605551275 and -3.605551275.

Subproblem 1

x + -5 = 3.605551275 Simplifying x + -5 = 3.605551275 Reorder the terms: -5 + x = 3.605551275 Solving -5 + x = 3.605551275 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 = 3.605551275 + 5 Combine like terms: -5 + 5 = 0 0 + x = 3.605551275 + 5 x = 3.605551275 + 5 Combine like terms: 3.605551275 + 5 = 8.605551275 x = 8.605551275 Simplifying x = 8.605551275

Subproblem 2

x + -5 = -3.605551275 Simplifying x + -5 = -3.605551275 Reorder the terms: -5 + x = -3.605551275 Solving -5 + x = -3.605551275 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 = -3.605551275 + 5 Combine like terms: -5 + 5 = 0 0 + x = -3.605551275 + 5 x = -3.605551275 + 5 Combine like terms: -3.605551275 + 5 = 1.394448725 x = 1.394448725 Simplifying x = 1.394448725

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.605551275, 1.394448725}

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