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

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


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

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

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

Multiply (5 + 2x) * (-3 + x)
x = (5(-3 + x) + 2x * (-3 + x))
x = ((-3 * 5 + x * 5) + 2x * (-3 + x))
x = ((-15 + 5x) + 2x * (-3 + x))
x = (-15 + 5x + (-3 * 2x + x * 2x))
x = (-15 + 5x + (-6x + 2x2))

Combine like terms: 5x + -6x = -1x
x = (-15 + -1x + 2x2)

Solving
x = -15 + -1x + 2x2

Solving for variable 'x'.

Reorder the terms:
15 + x + x + -2x2 = -15 + -1x + 2x2 + 15 + x + -2x2

Combine like terms: x + x = 2x
15 + 2x + -2x2 = -15 + -1x + 2x2 + 15 + x + -2x2

Reorder the terms:
15 + 2x + -2x2 = -15 + 15 + -1x + x + 2x2 + -2x2

Combine like terms: -15 + 15 = 0
15 + 2x + -2x2 = 0 + -1x + x + 2x2 + -2x2
15 + 2x + -2x2 = -1x + x + 2x2 + -2x2

Combine like terms: -1x + x = 0
15 + 2x + -2x2 = 0 + 2x2 + -2x2
15 + 2x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
15 + 2x + -2x2 = 0

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

Divide each side by '-2'.
-7.5 + -1x + x2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + -1x + 7.5 + x2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + -1x + x2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + -1x + x2 = 0 + 7.5
-1x + x2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
-1x + x2 = 7.5

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 7.5 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 7.5 + 0.25

Combine like terms: 7.5 + 0.25 = 7.75
0.25 + -1x + x2 = 7.75

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 7.75

Calculate the square root of the right side: 2.783882181

Break this problem into two subproblems by setting 
(x + -0.5) equal to 2.783882181 and -2.783882181.

Subproblem 1

x + -0.5 = 2.783882181 Simplifying x + -0.5 = 2.783882181 Reorder the terms: -0.5 + x = 2.783882181 Solving -0.5 + x = 2.783882181 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 2.783882181 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 2.783882181 + 0.5 x = 2.783882181 + 0.5 Combine like terms: 2.783882181 + 0.5 = 3.283882181 x = 3.283882181 Simplifying x = 3.283882181

Subproblem 2

x + -0.5 = -2.783882181 Simplifying x + -0.5 = -2.783882181 Reorder the terms: -0.5 + x = -2.783882181 Solving -0.5 + x = -2.783882181 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -2.783882181 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -2.783882181 + 0.5 x = -2.783882181 + 0.5 Combine like terms: -2.783882181 + 0.5 = -2.283882181 x = -2.283882181 Simplifying x = -2.283882181

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.283882181, -2.283882181}

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