5x(3-x)=7-x

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Solution for 5x(3-x)=7-x equation:


Simplifying
5x(3 + -1x) = 7 + -1x
(3 * 5x + -1x * 5x) = 7 + -1x
(15x + -5x2) = 7 + -1x

Solving
15x + -5x2 = 7 + -1x

Solving for variable 'x'.

Reorder the terms:
-7 + 15x + x + -5x2 = 7 + -1x + -7 + x

Combine like terms: 15x + x = 16x
-7 + 16x + -5x2 = 7 + -1x + -7 + x

Reorder the terms:
-7 + 16x + -5x2 = 7 + -7 + -1x + x

Combine like terms: 7 + -7 = 0
-7 + 16x + -5x2 = 0 + -1x + x
-7 + 16x + -5x2 = -1x + x

Combine like terms: -1x + x = 0
-7 + 16x + -5x2 = 0

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
1.4 + -3.2x + x2 = 0

Move the constant term to the right:

Add '-1.4' to each side of the equation.
1.4 + -3.2x + -1.4 + x2 = 0 + -1.4

Reorder the terms:
1.4 + -1.4 + -3.2x + x2 = 0 + -1.4

Combine like terms: 1.4 + -1.4 = 0.0
0.0 + -3.2x + x2 = 0 + -1.4
-3.2x + x2 = 0 + -1.4

Combine like terms: 0 + -1.4 = -1.4
-3.2x + x2 = -1.4

The x term is -3.2x.  Take half its coefficient (-1.6).
Square it (2.56) and add it to both sides.

Add '2.56' to each side of the equation.
-3.2x + 2.56 + x2 = -1.4 + 2.56

Reorder the terms:
2.56 + -3.2x + x2 = -1.4 + 2.56

Combine like terms: -1.4 + 2.56 = 1.16
2.56 + -3.2x + x2 = 1.16

Factor a perfect square on the left side:
(x + -1.6)(x + -1.6) = 1.16

Calculate the square root of the right side: 1.077032961

Break this problem into two subproblems by setting 
(x + -1.6) equal to 1.077032961 and -1.077032961.

Subproblem 1

x + -1.6 = 1.077032961 Simplifying x + -1.6 = 1.077032961 Reorder the terms: -1.6 + x = 1.077032961 Solving -1.6 + x = 1.077032961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.6' to each side of the equation. -1.6 + 1.6 + x = 1.077032961 + 1.6 Combine like terms: -1.6 + 1.6 = 0.0 0.0 + x = 1.077032961 + 1.6 x = 1.077032961 + 1.6 Combine like terms: 1.077032961 + 1.6 = 2.677032961 x = 2.677032961 Simplifying x = 2.677032961

Subproblem 2

x + -1.6 = -1.077032961 Simplifying x + -1.6 = -1.077032961 Reorder the terms: -1.6 + x = -1.077032961 Solving -1.6 + x = -1.077032961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.6' to each side of the equation. -1.6 + 1.6 + x = -1.077032961 + 1.6 Combine like terms: -1.6 + 1.6 = 0.0 0.0 + x = -1.077032961 + 1.6 x = -1.077032961 + 1.6 Combine like terms: -1.077032961 + 1.6 = 0.522967039 x = 0.522967039 Simplifying x = 0.522967039

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.677032961, 0.522967039}

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