4x(2x-7)=112

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Solution for 4x(2x-7)=112 equation: 


Simplifying
4x(2x + -7) = 112

Reorder the terms:
4x(-7 + 2x) = 112
(-7 * 4x + 2x * 4x) = 112
(-28x + 8x2) = 112

Solving
-28x + 8x2 = 112

Solving for variable 'x'.

Reorder the terms:
-112 + -28x + 8x2 = 112 + -112

Combine like terms: 112 + -112 = 0
-112 + -28x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-28 + -7x + 2x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-28 + -7x + 2x2)' equal to zero and attempt to solve:

Simplifying
-28 + -7x + 2x2 = 0

Solving
-28 + -7x + 2x2 = 0

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

Divide each side by '2'.
-14 + -3.5x + x2 = 0

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + -3.5x + 14 + x2 = 0 + 14

Reorder the terms:
-14 + 14 + -3.5x + x2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + -3.5x + x2 = 0 + 14
-3.5x + x2 = 0 + 14

Combine like terms: 0 + 14 = 14
-3.5x + x2 = 14

The x term is -3.5x.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5x + 3.0625 + x2 = 14 + 3.0625

Reorder the terms:
3.0625 + -3.5x + x2 = 14 + 3.0625

Combine like terms: 14 + 3.0625 = 17.0625
3.0625 + -3.5x + x2 = 17.0625

Factor a perfect square on the left side:
(x + -1.75)(x + -1.75) = 17.0625

Calculate the square root of the right side: 4.13067791

Break this problem into two subproblems by setting 
(x + -1.75) equal to 4.13067791 and -4.13067791.


Subproblem 1
x + -1.75 = 4.13067791

Simplifying
x + -1.75 = 4.13067791

Reorder the terms:
-1.75 + x = 4.13067791

Solving
-1.75 + x = 4.13067791

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.75' to each side of the equation.
-1.75 + 1.75 + x = 4.13067791 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + x = 4.13067791 + 1.75
x = 4.13067791 + 1.75

Combine like terms: 4.13067791 + 1.75 = 5.88067791
x = 5.88067791

Simplifying
x = 5.88067791


Subproblem 2
x + -1.75 = -4.13067791

Simplifying
x + -1.75 = -4.13067791

Reorder the terms:
-1.75 + x = -4.13067791

Solving
-1.75 + x = -4.13067791

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.75' to each side of the equation.
-1.75 + 1.75 + x = -4.13067791 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + x = -4.13067791 + 1.75
x = -4.13067791 + 1.75

Combine like terms: -4.13067791 + 1.75 = -2.38067791
x = -2.38067791

Simplifying
x = -2.38067791


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.88067791, -2.38067791}
Solution
x = {5.88067791, -2.38067791}

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