(5+x)(10+x)=54

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Solution for (5+x)(10+x)=54 equation:


Simplifying
(5 + x)(10 + x) = 54

Multiply (5 + x) * (10 + x)
(5(10 + x) + x(10 + x)) = 54
((10 * 5 + x * 5) + x(10 + x)) = 54
((50 + 5x) + x(10 + x)) = 54
(50 + 5x + (10 * x + x * x)) = 54
(50 + 5x + (10x + x2)) = 54

Combine like terms: 5x + 10x = 15x
(50 + 15x + x2) = 54

Solving
50 + 15x + x2 = 54

Solving for variable 'x'.

Reorder the terms:
50 + -54 + 15x + x2 = 54 + -54

Combine like terms: 50 + -54 = -4
-4 + 15x + x2 = 54 + -54

Combine like terms: 54 + -54 = 0
-4 + 15x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 15x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + 15x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 15x + x2 = 0 + 4
15x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
15x + x2 = 4

The x term is 15x.  Take half its coefficient (7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
15x + 56.25 + x2 = 4 + 56.25

Reorder the terms:
56.25 + 15x + x2 = 4 + 56.25

Combine like terms: 4 + 56.25 = 60.25
56.25 + 15x + x2 = 60.25

Factor a perfect square on the left side:
(x + 7.5)(x + 7.5) = 60.25

Calculate the square root of the right side: 7.762087348

Break this problem into two subproblems by setting 
(x + 7.5) equal to 7.762087348 and -7.762087348.

Subproblem 1

x + 7.5 = 7.762087348 Simplifying x + 7.5 = 7.762087348 Reorder the terms: 7.5 + x = 7.762087348 Solving 7.5 + x = 7.762087348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = 7.762087348 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = 7.762087348 + -7.5 x = 7.762087348 + -7.5 Combine like terms: 7.762087348 + -7.5 = 0.262087348 x = 0.262087348 Simplifying x = 0.262087348

Subproblem 2

x + 7.5 = -7.762087348 Simplifying x + 7.5 = -7.762087348 Reorder the terms: 7.5 + x = -7.762087348 Solving 7.5 + x = -7.762087348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = -7.762087348 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = -7.762087348 + -7.5 x = -7.762087348 + -7.5 Combine like terms: -7.762087348 + -7.5 = -15.262087348 x = -15.262087348 Simplifying x = -15.262087348

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.262087348, -15.262087348}

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