x(x+14)=(x+7)2

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Solution for x(x+14)=(x+7)2 equation:


Simplifying
x(x + 14) = (x + 7) * 2

Reorder the terms:
x(14 + x) = (x + 7) * 2
(14 * x + x * x) = (x + 7) * 2
(14x + x2) = (x + 7) * 2

Reorder the terms:
14x + x2 = (7 + x) * 2

Reorder the terms for easier multiplication:
14x + x2 = 2(7 + x)
14x + x2 = (7 * 2 + x * 2)
14x + x2 = (14 + 2x)

Solving
14x + x2 = 14 + 2x

Solving for variable 'x'.

Reorder the terms:
-14 + 14x + -2x + x2 = 14 + 2x + -14 + -2x

Combine like terms: 14x + -2x = 12x
-14 + 12x + x2 = 14 + 2x + -14 + -2x

Reorder the terms:
-14 + 12x + x2 = 14 + -14 + 2x + -2x

Combine like terms: 14 + -14 = 0
-14 + 12x + x2 = 0 + 2x + -2x
-14 + 12x + x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-14 + 12x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + 12x + 14 + x2 = 0 + 14

Reorder the terms:
-14 + 14 + 12x + x2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + 12x + x2 = 0 + 14
12x + x2 = 0 + 14

Combine like terms: 0 + 14 = 14
12x + x2 = 14

The x term is 12x.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12x + 36 + x2 = 14 + 36

Reorder the terms:
36 + 12x + x2 = 14 + 36

Combine like terms: 14 + 36 = 50
36 + 12x + x2 = 50

Factor a perfect square on the left side:
(x + 6)(x + 6) = 50

Calculate the square root of the right side: 7.071067812

Break this problem into two subproblems by setting 
(x + 6) equal to 7.071067812 and -7.071067812.

Subproblem 1

x + 6 = 7.071067812 Simplifying x + 6 = 7.071067812 Reorder the terms: 6 + x = 7.071067812 Solving 6 + x = 7.071067812 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + x = 7.071067812 + -6 Combine like terms: 6 + -6 = 0 0 + x = 7.071067812 + -6 x = 7.071067812 + -6 Combine like terms: 7.071067812 + -6 = 1.071067812 x = 1.071067812 Simplifying x = 1.071067812

Subproblem 2

x + 6 = -7.071067812 Simplifying x + 6 = -7.071067812 Reorder the terms: 6 + x = -7.071067812 Solving 6 + x = -7.071067812 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + x = -7.071067812 + -6 Combine like terms: 6 + -6 = 0 0 + x = -7.071067812 + -6 x = -7.071067812 + -6 Combine like terms: -7.071067812 + -6 = -13.071067812 x = -13.071067812 Simplifying x = -13.071067812

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.071067812, -13.071067812}

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