(x-3)(2x+1)=5(x-2)

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Solution for (x-3)(2x+1)=5(x-2) equation:


Simplifying
(x + -3)(2x + 1) = 5(x + -2)

Reorder the terms:
(-3 + x)(2x + 1) = 5(x + -2)

Reorder the terms:
(-3 + x)(1 + 2x) = 5(x + -2)

Multiply (-3 + x) * (1 + 2x)
(-3(1 + 2x) + x(1 + 2x)) = 5(x + -2)
((1 * -3 + 2x * -3) + x(1 + 2x)) = 5(x + -2)
((-3 + -6x) + x(1 + 2x)) = 5(x + -2)
(-3 + -6x + (1 * x + 2x * x)) = 5(x + -2)
(-3 + -6x + (1x + 2x2)) = 5(x + -2)

Combine like terms: -6x + 1x = -5x
(-3 + -5x + 2x2) = 5(x + -2)

Reorder the terms:
-3 + -5x + 2x2 = 5(-2 + x)
-3 + -5x + 2x2 = (-2 * 5 + x * 5)
-3 + -5x + 2x2 = (-10 + 5x)

Solving
-3 + -5x + 2x2 = -10 + 5x

Solving for variable 'x'.

Reorder the terms:
-3 + 10 + -5x + -5x + 2x2 = -10 + 5x + 10 + -5x

Combine like terms: -3 + 10 = 7
7 + -5x + -5x + 2x2 = -10 + 5x + 10 + -5x

Combine like terms: -5x + -5x = -10x
7 + -10x + 2x2 = -10 + 5x + 10 + -5x

Reorder the terms:
7 + -10x + 2x2 = -10 + 10 + 5x + -5x

Combine like terms: -10 + 10 = 0
7 + -10x + 2x2 = 0 + 5x + -5x
7 + -10x + 2x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
7 + -10x + 2x2 = 0

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

Divide each side by '2'.
3.5 + -5x + x2 = 0

Move the constant term to the right:

Add '-3.5' to each side of the equation.
3.5 + -5x + -3.5 + x2 = 0 + -3.5

Reorder the terms:
3.5 + -3.5 + -5x + x2 = 0 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + -5x + x2 = 0 + -3.5
-5x + x2 = 0 + -3.5

Combine like terms: 0 + -3.5 = -3.5
-5x + x2 = -3.5

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -3.5 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -3.5 + 6.25

Combine like terms: -3.5 + 6.25 = 2.75
6.25 + -5x + x2 = 2.75

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(x + -2.5) equal to 1.658312395 and -1.658312395.

Subproblem 1

x + -2.5 = 1.658312395 Simplifying x + -2.5 = 1.658312395 Reorder the terms: -2.5 + x = 1.658312395 Solving -2.5 + x = 1.658312395 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 1.658312395 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 1.658312395 + 2.5 x = 1.658312395 + 2.5 Combine like terms: 1.658312395 + 2.5 = 4.158312395 x = 4.158312395 Simplifying x = 4.158312395

Subproblem 2

x + -2.5 = -1.658312395 Simplifying x + -2.5 = -1.658312395 Reorder the terms: -2.5 + x = -1.658312395 Solving -2.5 + x = -1.658312395 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -1.658312395 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -1.658312395 + 2.5 x = -1.658312395 + 2.5 Combine like terms: -1.658312395 + 2.5 = 0.841687605 x = 0.841687605 Simplifying x = 0.841687605

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.158312395, 0.841687605}

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