16v^2+24v+9=360

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Solution for 16v^2+24v+9=360 equation:


Simplifying
16v2 + 24v + 9 = 360

Reorder the terms:
9 + 24v + 16v2 = 360

Solving
9 + 24v + 16v2 = 360

Solving for variable 'v'.

Reorder the terms:
9 + -360 + 24v + 16v2 = 360 + -360

Combine like terms: 9 + -360 = -351
-351 + 24v + 16v2 = 360 + -360

Combine like terms: 360 + -360 = 0
-351 + 24v + 16v2 = 0

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

Divide each side by '16'.
-21.9375 + 1.5v + v2 = 0

Move the constant term to the right:

Add '21.9375' to each side of the equation.
-21.9375 + 1.5v + 21.9375 + v2 = 0 + 21.9375

Reorder the terms:
-21.9375 + 21.9375 + 1.5v + v2 = 0 + 21.9375

Combine like terms: -21.9375 + 21.9375 = 0.0000
0.0000 + 1.5v + v2 = 0 + 21.9375
1.5v + v2 = 0 + 21.9375

Combine like terms: 0 + 21.9375 = 21.9375
1.5v + v2 = 21.9375

The v term is 1.5v.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5v + 0.5625 + v2 = 21.9375 + 0.5625

Reorder the terms:
0.5625 + 1.5v + v2 = 21.9375 + 0.5625

Combine like terms: 21.9375 + 0.5625 = 22.5
0.5625 + 1.5v + v2 = 22.5

Factor a perfect square on the left side:
(v + 0.75)(v + 0.75) = 22.5

Calculate the square root of the right side: 4.74341649

Break this problem into two subproblems by setting 
(v + 0.75) equal to 4.74341649 and -4.74341649.

Subproblem 1

v + 0.75 = 4.74341649 Simplifying v + 0.75 = 4.74341649 Reorder the terms: 0.75 + v = 4.74341649 Solving 0.75 + v = 4.74341649 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + v = 4.74341649 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + v = 4.74341649 + -0.75 v = 4.74341649 + -0.75 Combine like terms: 4.74341649 + -0.75 = 3.99341649 v = 3.99341649 Simplifying v = 3.99341649

Subproblem 2

v + 0.75 = -4.74341649 Simplifying v + 0.75 = -4.74341649 Reorder the terms: 0.75 + v = -4.74341649 Solving 0.75 + v = -4.74341649 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + v = -4.74341649 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + v = -4.74341649 + -0.75 v = -4.74341649 + -0.75 Combine like terms: -4.74341649 + -0.75 = -5.49341649 v = -5.49341649 Simplifying v = -5.49341649

Solution

The solution to the problem is based on the solutions from the subproblems. v = {3.99341649, -5.49341649}

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