3v^2+5v-18=0

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Solution for 3v^2+5v-18=0 equation:


Simplifying
3v2 + 5v + -18 = 0

Reorder the terms:
-18 + 5v + 3v2 = 0

Solving
-18 + 5v + 3v2 = 0

Solving for variable 'v'.

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

Divide each side by '3'.
-6 + 1.666666667v + v2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 1.666666667v + 6 + v2 = 0 + 6

Reorder the terms:
-6 + 6 + 1.666666667v + v2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 1.666666667v + v2 = 0 + 6
1.666666667v + v2 = 0 + 6

Combine like terms: 0 + 6 = 6
1.666666667v + v2 = 6

The v term is 1.666666667v.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667v + 0.6944444447 + v2 = 6 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667v + v2 = 6 + 0.6944444447

Combine like terms: 6 + 0.6944444447 = 6.6944444447
0.6944444447 + 1.666666667v + v2 = 6.6944444447

Factor a perfect square on the left side:
(v + 0.8333333335)(v + 0.8333333335) = 6.6944444447

Calculate the square root of the right side: 2.587362449

Break this problem into two subproblems by setting 
(v + 0.8333333335) equal to 2.587362449 and -2.587362449.

Subproblem 1

v + 0.8333333335 = 2.587362449 Simplifying v + 0.8333333335 = 2.587362449 Reorder the terms: 0.8333333335 + v = 2.587362449 Solving 0.8333333335 + v = 2.587362449 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + v = 2.587362449 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + v = 2.587362449 + -0.8333333335 v = 2.587362449 + -0.8333333335 Combine like terms: 2.587362449 + -0.8333333335 = 1.7540291155 v = 1.7540291155 Simplifying v = 1.7540291155

Subproblem 2

v + 0.8333333335 = -2.587362449 Simplifying v + 0.8333333335 = -2.587362449 Reorder the terms: 0.8333333335 + v = -2.587362449 Solving 0.8333333335 + v = -2.587362449 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + v = -2.587362449 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + v = -2.587362449 + -0.8333333335 v = -2.587362449 + -0.8333333335 Combine like terms: -2.587362449 + -0.8333333335 = -3.4206957825 v = -3.4206957825 Simplifying v = -3.4206957825

Solution

The solution to the problem is based on the solutions from the subproblems. v = {1.7540291155, -3.4206957825}

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