v^2+4v-78=-7

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Solution for v^2+4v-78=-7 equation:


Simplifying
v2 + 4v + -78 = -7

Reorder the terms:
-78 + 4v + v2 = -7

Solving
-78 + 4v + v2 = -7

Solving for variable 'v'.

Reorder the terms:
-78 + 7 + 4v + v2 = -7 + 7

Combine like terms: -78 + 7 = -71
-71 + 4v + v2 = -7 + 7

Combine like terms: -7 + 7 = 0
-71 + 4v + v2 = 0

Begin completing the square.

Move the constant term to the right:

Add '71' to each side of the equation.
-71 + 4v + 71 + v2 = 0 + 71

Reorder the terms:
-71 + 71 + 4v + v2 = 0 + 71

Combine like terms: -71 + 71 = 0
0 + 4v + v2 = 0 + 71
4v + v2 = 0 + 71

Combine like terms: 0 + 71 = 71
4v + v2 = 71

The v term is 4v.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4v + 4 + v2 = 71 + 4

Reorder the terms:
4 + 4v + v2 = 71 + 4

Combine like terms: 71 + 4 = 75
4 + 4v + v2 = 75

Factor a perfect square on the left side:
(v + 2)(v + 2) = 75

Calculate the square root of the right side: 8.660254038

Break this problem into two subproblems by setting 
(v + 2) equal to 8.660254038 and -8.660254038.

Subproblem 1

v + 2 = 8.660254038 Simplifying v + 2 = 8.660254038 Reorder the terms: 2 + v = 8.660254038 Solving 2 + v = 8.660254038 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + v = 8.660254038 + -2 Combine like terms: 2 + -2 = 0 0 + v = 8.660254038 + -2 v = 8.660254038 + -2 Combine like terms: 8.660254038 + -2 = 6.660254038 v = 6.660254038 Simplifying v = 6.660254038

Subproblem 2

v + 2 = -8.660254038 Simplifying v + 2 = -8.660254038 Reorder the terms: 2 + v = -8.660254038 Solving 2 + v = -8.660254038 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + v = -8.660254038 + -2 Combine like terms: 2 + -2 = 0 0 + v = -8.660254038 + -2 v = -8.660254038 + -2 Combine like terms: -8.660254038 + -2 = -10.660254038 v = -10.660254038 Simplifying v = -10.660254038

Solution

The solution to the problem is based on the solutions from the subproblems. v = {6.660254038, -10.660254038}

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