2v^2+5v-12=v^2+16

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Solution for 2v^2+5v-12=v^2+16 equation:


Simplifying
2v2 + 5v + -12 = v2 + 16

Reorder the terms:
-12 + 5v + 2v2 = v2 + 16

Reorder the terms:
-12 + 5v + 2v2 = 16 + v2

Solving
-12 + 5v + 2v2 = 16 + v2

Solving for variable 'v'.

Reorder the terms:
-12 + -16 + 5v + 2v2 + -1v2 = 16 + v2 + -16 + -1v2

Combine like terms: -12 + -16 = -28
-28 + 5v + 2v2 + -1v2 = 16 + v2 + -16 + -1v2

Combine like terms: 2v2 + -1v2 = 1v2
-28 + 5v + 1v2 = 16 + v2 + -16 + -1v2

Reorder the terms:
-28 + 5v + 1v2 = 16 + -16 + v2 + -1v2

Combine like terms: 16 + -16 = 0
-28 + 5v + 1v2 = 0 + v2 + -1v2
-28 + 5v + 1v2 = v2 + -1v2

Combine like terms: v2 + -1v2 = 0
-28 + 5v + 1v2 = 0

Begin completing the square.

Move the constant term to the right:

Add '28' to each side of the equation.
-28 + 5v + 28 + v2 = 0 + 28

Reorder the terms:
-28 + 28 + 5v + v2 = 0 + 28

Combine like terms: -28 + 28 = 0
0 + 5v + v2 = 0 + 28
5v + v2 = 0 + 28

Combine like terms: 0 + 28 = 28
5v + v2 = 28

The v term is 5v.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5v + 6.25 + v2 = 28 + 6.25

Reorder the terms:
6.25 + 5v + v2 = 28 + 6.25

Combine like terms: 28 + 6.25 = 34.25
6.25 + 5v + v2 = 34.25

Factor a perfect square on the left side:
(v + 2.5)(v + 2.5) = 34.25

Calculate the square root of the right side: 5.852349955

Break this problem into two subproblems by setting 
(v + 2.5) equal to 5.852349955 and -5.852349955.

Subproblem 1

v + 2.5 = 5.852349955 Simplifying v + 2.5 = 5.852349955 Reorder the terms: 2.5 + v = 5.852349955 Solving 2.5 + v = 5.852349955 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + v = 5.852349955 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + v = 5.852349955 + -2.5 v = 5.852349955 + -2.5 Combine like terms: 5.852349955 + -2.5 = 3.352349955 v = 3.352349955 Simplifying v = 3.352349955

Subproblem 2

v + 2.5 = -5.852349955 Simplifying v + 2.5 = -5.852349955 Reorder the terms: 2.5 + v = -5.852349955 Solving 2.5 + v = -5.852349955 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + v = -5.852349955 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + v = -5.852349955 + -2.5 v = -5.852349955 + -2.5 Combine like terms: -5.852349955 + -2.5 = -8.352349955 v = -8.352349955 Simplifying v = -8.352349955

Solution

The solution to the problem is based on the solutions from the subproblems. v = {3.352349955, -8.352349955}

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