(2x+3)(x+7)=1600

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Solution for (2x+3)(x+7)=1600 equation:


Simplifying
(2x + 3)(x + 7) = 1600

Reorder the terms:
(3 + 2x)(x + 7) = 1600

Reorder the terms:
(3 + 2x)(7 + x) = 1600

Multiply (3 + 2x) * (7 + x)
(3(7 + x) + 2x * (7 + x)) = 1600
((7 * 3 + x * 3) + 2x * (7 + x)) = 1600
((21 + 3x) + 2x * (7 + x)) = 1600
(21 + 3x + (7 * 2x + x * 2x)) = 1600
(21 + 3x + (14x + 2x2)) = 1600

Combine like terms: 3x + 14x = 17x
(21 + 17x + 2x2) = 1600

Solving
21 + 17x + 2x2 = 1600

Solving for variable 'x'.

Reorder the terms:
21 + -1600 + 17x + 2x2 = 1600 + -1600

Combine like terms: 21 + -1600 = -1579
-1579 + 17x + 2x2 = 1600 + -1600

Combine like terms: 1600 + -1600 = 0
-1579 + 17x + 2x2 = 0

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

Divide each side by '2'.
-789.5 + 8.5x + x2 = 0

Move the constant term to the right:

Add '789.5' to each side of the equation.
-789.5 + 8.5x + 789.5 + x2 = 0 + 789.5

Reorder the terms:
-789.5 + 789.5 + 8.5x + x2 = 0 + 789.5

Combine like terms: -789.5 + 789.5 = 0.0
0.0 + 8.5x + x2 = 0 + 789.5
8.5x + x2 = 0 + 789.5

Combine like terms: 0 + 789.5 = 789.5
8.5x + x2 = 789.5

The x term is 8.5x.  Take half its coefficient (4.25).
Square it (18.0625) and add it to both sides.

Add '18.0625' to each side of the equation.
8.5x + 18.0625 + x2 = 789.5 + 18.0625

Reorder the terms:
18.0625 + 8.5x + x2 = 789.5 + 18.0625

Combine like terms: 789.5 + 18.0625 = 807.5625
18.0625 + 8.5x + x2 = 807.5625

Factor a perfect square on the left side:
(x + 4.25)(x + 4.25) = 807.5625

Calculate the square root of the right side: 28.417644167

Break this problem into two subproblems by setting 
(x + 4.25) equal to 28.417644167 and -28.417644167.

Subproblem 1

x + 4.25 = 28.417644167 Simplifying x + 4.25 = 28.417644167 Reorder the terms: 4.25 + x = 28.417644167 Solving 4.25 + x = 28.417644167 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = 28.417644167 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = 28.417644167 + -4.25 x = 28.417644167 + -4.25 Combine like terms: 28.417644167 + -4.25 = 24.167644167 x = 24.167644167 Simplifying x = 24.167644167

Subproblem 2

x + 4.25 = -28.417644167 Simplifying x + 4.25 = -28.417644167 Reorder the terms: 4.25 + x = -28.417644167 Solving 4.25 + x = -28.417644167 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = -28.417644167 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = -28.417644167 + -4.25 x = -28.417644167 + -4.25 Combine like terms: -28.417644167 + -4.25 = -32.667644167 x = -32.667644167 Simplifying x = -32.667644167

Solution

The solution to the problem is based on the solutions from the subproblems. x = {24.167644167, -32.667644167}

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