2x+7=2x^2-16

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Solution for 2x+7=2x^2-16 equation:


Simplifying
2x + 7 = 2x2 + -16

Reorder the terms:
7 + 2x = 2x2 + -16

Reorder the terms:
7 + 2x = -16 + 2x2

Solving
7 + 2x = -16 + 2x2

Solving for variable 'x'.

Reorder the terms:
7 + 16 + 2x + -2x2 = -16 + 2x2 + 16 + -2x2

Combine like terms: 7 + 16 = 23
23 + 2x + -2x2 = -16 + 2x2 + 16 + -2x2

Reorder the terms:
23 + 2x + -2x2 = -16 + 16 + 2x2 + -2x2

Combine like terms: -16 + 16 = 0
23 + 2x + -2x2 = 0 + 2x2 + -2x2
23 + 2x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
23 + 2x + -2x2 = 0

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

Divide each side by '-2'.
-11.5 + -1x + x2 = 0

Move the constant term to the right:

Add '11.5' to each side of the equation.
-11.5 + -1x + 11.5 + x2 = 0 + 11.5

Reorder the terms:
-11.5 + 11.5 + -1x + x2 = 0 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + -1x + x2 = 0 + 11.5
-1x + x2 = 0 + 11.5

Combine like terms: 0 + 11.5 = 11.5
-1x + x2 = 11.5

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 11.5 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 11.5 + 0.25

Combine like terms: 11.5 + 0.25 = 11.75
0.25 + -1x + x2 = 11.75

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 11.75

Calculate the square root of the right side: 3.4278273

Break this problem into two subproblems by setting 
(x + -0.5) equal to 3.4278273 and -3.4278273.

Subproblem 1

x + -0.5 = 3.4278273 Simplifying x + -0.5 = 3.4278273 Reorder the terms: -0.5 + x = 3.4278273 Solving -0.5 + x = 3.4278273 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 3.4278273 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 3.4278273 + 0.5 x = 3.4278273 + 0.5 Combine like terms: 3.4278273 + 0.5 = 3.9278273 x = 3.9278273 Simplifying x = 3.9278273

Subproblem 2

x + -0.5 = -3.4278273 Simplifying x + -0.5 = -3.4278273 Reorder the terms: -0.5 + x = -3.4278273 Solving -0.5 + x = -3.4278273 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -3.4278273 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -3.4278273 + 0.5 x = -3.4278273 + 0.5 Combine like terms: -3.4278273 + 0.5 = -2.9278273 x = -2.9278273 Simplifying x = -2.9278273

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.9278273, -2.9278273}

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