7x=(4x+9)(x-1)

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Solution for 7x=(4x+9)(x-1) equation:


Simplifying
7x = (4x + 9)(x + -1)

Reorder the terms:
7x = (9 + 4x)(x + -1)

Reorder the terms:
7x = (9 + 4x)(-1 + x)

Multiply (9 + 4x) * (-1 + x)
7x = (9(-1 + x) + 4x * (-1 + x))
7x = ((-1 * 9 + x * 9) + 4x * (-1 + x))
7x = ((-9 + 9x) + 4x * (-1 + x))
7x = (-9 + 9x + (-1 * 4x + x * 4x))
7x = (-9 + 9x + (-4x + 4x2))

Combine like terms: 9x + -4x = 5x
7x = (-9 + 5x + 4x2)

Solving
7x = -9 + 5x + 4x2

Solving for variable 'x'.

Reorder the terms:
9 + 7x + -5x + -4x2 = -9 + 5x + 4x2 + 9 + -5x + -4x2

Combine like terms: 7x + -5x = 2x
9 + 2x + -4x2 = -9 + 5x + 4x2 + 9 + -5x + -4x2

Reorder the terms:
9 + 2x + -4x2 = -9 + 9 + 5x + -5x + 4x2 + -4x2

Combine like terms: -9 + 9 = 0
9 + 2x + -4x2 = 0 + 5x + -5x + 4x2 + -4x2
9 + 2x + -4x2 = 5x + -5x + 4x2 + -4x2

Combine like terms: 5x + -5x = 0
9 + 2x + -4x2 = 0 + 4x2 + -4x2
9 + 2x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
9 + 2x + -4x2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-2.25 + -0.5x + x2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + -0.5x + 2.25 + x2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + -0.5x + x2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + -0.5x + x2 = 0 + 2.25
-0.5x + x2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
-0.5x + x2 = 2.25

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 2.25 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 2.25 + 0.0625

Combine like terms: 2.25 + 0.0625 = 2.3125
0.0625 + -0.5x + x2 = 2.3125

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 2.3125

Calculate the square root of the right side: 1.520690633

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.520690633 and -1.520690633.

Subproblem 1

x + -0.25 = 1.520690633 Simplifying x + -0.25 = 1.520690633 Reorder the terms: -0.25 + x = 1.520690633 Solving -0.25 + x = 1.520690633 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 1.520690633 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 1.520690633 + 0.25 x = 1.520690633 + 0.25 Combine like terms: 1.520690633 + 0.25 = 1.770690633 x = 1.770690633 Simplifying x = 1.770690633

Subproblem 2

x + -0.25 = -1.520690633 Simplifying x + -0.25 = -1.520690633 Reorder the terms: -0.25 + x = -1.520690633 Solving -0.25 + x = -1.520690633 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -1.520690633 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -1.520690633 + 0.25 x = -1.520690633 + 0.25 Combine like terms: -1.520690633 + 0.25 = -1.270690633 x = -1.270690633 Simplifying x = -1.270690633

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.770690633, -1.270690633}

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