2x^2+14x=x^2+23

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Solution for 2x^2+14x=x^2+23 equation:


Simplifying
2x2 + 14x = x2 + 23

Reorder the terms:
14x + 2x2 = x2 + 23

Reorder the terms:
14x + 2x2 = 23 + x2

Solving
14x + 2x2 = 23 + x2

Solving for variable 'x'.

Reorder the terms:
-23 + 14x + 2x2 + -1x2 = 23 + x2 + -23 + -1x2

Combine like terms: 2x2 + -1x2 = 1x2
-23 + 14x + 1x2 = 23 + x2 + -23 + -1x2

Reorder the terms:
-23 + 14x + 1x2 = 23 + -23 + x2 + -1x2

Combine like terms: 23 + -23 = 0
-23 + 14x + 1x2 = 0 + x2 + -1x2
-23 + 14x + 1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-23 + 14x + 1x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + 14x + 23 + x2 = 0 + 23

Reorder the terms:
-23 + 23 + 14x + x2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + 14x + x2 = 0 + 23
14x + x2 = 0 + 23

Combine like terms: 0 + 23 = 23
14x + x2 = 23

The x term is 14x.  Take half its coefficient (7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
14x + 49 + x2 = 23 + 49

Reorder the terms:
49 + 14x + x2 = 23 + 49

Combine like terms: 23 + 49 = 72
49 + 14x + x2 = 72

Factor a perfect square on the left side:
(x + 7)(x + 7) = 72

Calculate the square root of the right side: 8.485281374

Break this problem into two subproblems by setting 
(x + 7) equal to 8.485281374 and -8.485281374.

Subproblem 1

x + 7 = 8.485281374 Simplifying x + 7 = 8.485281374 Reorder the terms: 7 + x = 8.485281374 Solving 7 + x = 8.485281374 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + x = 8.485281374 + -7 Combine like terms: 7 + -7 = 0 0 + x = 8.485281374 + -7 x = 8.485281374 + -7 Combine like terms: 8.485281374 + -7 = 1.485281374 x = 1.485281374 Simplifying x = 1.485281374

Subproblem 2

x + 7 = -8.485281374 Simplifying x + 7 = -8.485281374 Reorder the terms: 7 + x = -8.485281374 Solving 7 + x = -8.485281374 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + x = -8.485281374 + -7 Combine like terms: 7 + -7 = 0 0 + x = -8.485281374 + -7 x = -8.485281374 + -7 Combine like terms: -8.485281374 + -7 = -15.485281374 x = -15.485281374 Simplifying x = -15.485281374

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.485281374, -15.485281374}

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