35v^2+19v-24=9

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Solution for 35v^2+19v-24=9 equation:


Simplifying
35v2 + 19v + -24 = 9

Reorder the terms:
-24 + 19v + 35v2 = 9

Solving
-24 + 19v + 35v2 = 9

Solving for variable 'v'.

Reorder the terms:
-24 + -9 + 19v + 35v2 = 9 + -9

Combine like terms: -24 + -9 = -33
-33 + 19v + 35v2 = 9 + -9

Combine like terms: 9 + -9 = 0
-33 + 19v + 35v2 = 0

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

Divide each side by '35'.
-0.9428571429 + 0.5428571429v + v2 = 0

Move the constant term to the right:

Add '0.9428571429' to each side of the equation.
-0.9428571429 + 0.5428571429v + 0.9428571429 + v2 = 0 + 0.9428571429

Reorder the terms:
-0.9428571429 + 0.9428571429 + 0.5428571429v + v2 = 0 + 0.9428571429

Combine like terms: -0.9428571429 + 0.9428571429 = 0.0000000000
0.0000000000 + 0.5428571429v + v2 = 0 + 0.9428571429
0.5428571429v + v2 = 0 + 0.9428571429

Combine like terms: 0 + 0.9428571429 = 0.9428571429
0.5428571429v + v2 = 0.9428571429

The v term is 0.5428571429v.  Take half its coefficient (0.2714285715).
Square it (0.07367346943) and add it to both sides.

Add '0.07367346943' to each side of the equation.
0.5428571429v + 0.07367346943 + v2 = 0.9428571429 + 0.07367346943

Reorder the terms:
0.07367346943 + 0.5428571429v + v2 = 0.9428571429 + 0.07367346943

Combine like terms: 0.9428571429 + 0.07367346943 = 1.01653061233
0.07367346943 + 0.5428571429v + v2 = 1.01653061233

Factor a perfect square on the left side:
(v + 0.2714285715)(v + 0.2714285715) = 1.01653061233

Calculate the square root of the right side: 1.008231428

Break this problem into two subproblems by setting 
(v + 0.2714285715) equal to 1.008231428 and -1.008231428.

Subproblem 1

v + 0.2714285715 = 1.008231428 Simplifying v + 0.2714285715 = 1.008231428 Reorder the terms: 0.2714285715 + v = 1.008231428 Solving 0.2714285715 + v = 1.008231428 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.2714285715' to each side of the equation. 0.2714285715 + -0.2714285715 + v = 1.008231428 + -0.2714285715 Combine like terms: 0.2714285715 + -0.2714285715 = 0.0000000000 0.0000000000 + v = 1.008231428 + -0.2714285715 v = 1.008231428 + -0.2714285715 Combine like terms: 1.008231428 + -0.2714285715 = 0.7368028565 v = 0.7368028565 Simplifying v = 0.7368028565

Subproblem 2

v + 0.2714285715 = -1.008231428 Simplifying v + 0.2714285715 = -1.008231428 Reorder the terms: 0.2714285715 + v = -1.008231428 Solving 0.2714285715 + v = -1.008231428 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-0.2714285715' to each side of the equation. 0.2714285715 + -0.2714285715 + v = -1.008231428 + -0.2714285715 Combine like terms: 0.2714285715 + -0.2714285715 = 0.0000000000 0.0000000000 + v = -1.008231428 + -0.2714285715 v = -1.008231428 + -0.2714285715 Combine like terms: -1.008231428 + -0.2714285715 = -1.2796599995 v = -1.2796599995 Simplifying v = -1.2796599995

Solution

The solution to the problem is based on the solutions from the subproblems. v = {0.7368028565, -1.2796599995}

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