10x-x^2=3x-10

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Solution for 10x-x^2=3x-10 equation:


Simplifying
10x + -1x2 = 3x + -10

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

Solving
10x + -1x2 = -10 + 3x

Solving for variable 'x'.

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

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

Reorder the terms:
10 + 7x + -1x2 = -10 + 10 + 3x + -3x

Combine like terms: -10 + 10 = 0
10 + 7x + -1x2 = 0 + 3x + -3x
10 + 7x + -1x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
10 + 7x + -1x2 = 0

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

Divide each side by '-1'.
-10 + -7x + x2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -7x + 10 + x2 = 0 + 10

Reorder the terms:
-10 + 10 + -7x + x2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -7x + x2 = 0 + 10
-7x + x2 = 0 + 10

Combine like terms: 0 + 10 = 10
-7x + x2 = 10

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = 10 + 12.25

Reorder the terms:
12.25 + -7x + x2 = 10 + 12.25

Combine like terms: 10 + 12.25 = 22.25
12.25 + -7x + x2 = 22.25

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 22.25

Calculate the square root of the right side: 4.716990566

Break this problem into two subproblems by setting 
(x + -3.5) equal to 4.716990566 and -4.716990566.

Subproblem 1

x + -3.5 = 4.716990566 Simplifying x + -3.5 = 4.716990566 Reorder the terms: -3.5 + x = 4.716990566 Solving -3.5 + x = 4.716990566 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 4.716990566 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 4.716990566 + 3.5 x = 4.716990566 + 3.5 Combine like terms: 4.716990566 + 3.5 = 8.216990566 x = 8.216990566 Simplifying x = 8.216990566

Subproblem 2

x + -3.5 = -4.716990566 Simplifying x + -3.5 = -4.716990566 Reorder the terms: -3.5 + x = -4.716990566 Solving -3.5 + x = -4.716990566 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -4.716990566 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -4.716990566 + 3.5 x = -4.716990566 + 3.5 Combine like terms: -4.716990566 + 3.5 = -1.216990566 x = -1.216990566 Simplifying x = -1.216990566

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.216990566, -1.216990566}

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