42v^2=36v+216

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Solution for 42v^2=36v+216 equation:


Simplifying
42v2 = 36v + 216

Reorder the terms:
42v2 = 216 + 36v

Solving
42v2 = 216 + 36v

Solving for variable 'v'.

Reorder the terms:
-216 + -36v + 42v2 = 216 + 36v + -216 + -36v

Reorder the terms:
-216 + -36v + 42v2 = 216 + -216 + 36v + -36v

Combine like terms: 216 + -216 = 0
-216 + -36v + 42v2 = 0 + 36v + -36v
-216 + -36v + 42v2 = 36v + -36v

Combine like terms: 36v + -36v = 0
-216 + -36v + 42v2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-36 + -6v + 7v2) = 0

Ignore the factor 6.

Subproblem 1

Set the factor '(-36 + -6v + 7v2)' equal to zero and attempt to solve: Simplifying -36 + -6v + 7v2 = 0 Solving -36 + -6v + 7v2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -5.142857143 + -0.8571428571v + v2 = 0 Move the constant term to the right: Add '5.142857143' to each side of the equation. -5.142857143 + -0.8571428571v + 5.142857143 + v2 = 0 + 5.142857143 Reorder the terms: -5.142857143 + 5.142857143 + -0.8571428571v + v2 = 0 + 5.142857143 Combine like terms: -5.142857143 + 5.142857143 = 0.000000000 0.000000000 + -0.8571428571v + v2 = 0 + 5.142857143 -0.8571428571v + v2 = 0 + 5.142857143 Combine like terms: 0 + 5.142857143 = 5.142857143 -0.8571428571v + v2 = 5.142857143 The v term is -0.8571428571v. Take half its coefficient (-0.4285714286). Square it (0.1836734694) and add it to both sides. Add '0.1836734694' to each side of the equation. -0.8571428571v + 0.1836734694 + v2 = 5.142857143 + 0.1836734694 Reorder the terms: 0.1836734694 + -0.8571428571v + v2 = 5.142857143 + 0.1836734694 Combine like terms: 5.142857143 + 0.1836734694 = 5.3265306124 0.1836734694 + -0.8571428571v + v2 = 5.3265306124 Factor a perfect square on the left side: (v + -0.4285714286)(v + -0.4285714286) = 5.3265306124 Calculate the square root of the right side: 2.307927775 Break this problem into two subproblems by setting (v + -0.4285714286) equal to 2.307927775 and -2.307927775.

Subproblem 1

v + -0.4285714286 = 2.307927775 Simplifying v + -0.4285714286 = 2.307927775 Reorder the terms: -0.4285714286 + v = 2.307927775 Solving -0.4285714286 + v = 2.307927775 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + v = 2.307927775 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + v = 2.307927775 + 0.4285714286 v = 2.307927775 + 0.4285714286 Combine like terms: 2.307927775 + 0.4285714286 = 2.7364992036 v = 2.7364992036 Simplifying v = 2.7364992036

Subproblem 2

v + -0.4285714286 = -2.307927775 Simplifying v + -0.4285714286 = -2.307927775 Reorder the terms: -0.4285714286 + v = -2.307927775 Solving -0.4285714286 + v = -2.307927775 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + v = -2.307927775 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + v = -2.307927775 + 0.4285714286 v = -2.307927775 + 0.4285714286 Combine like terms: -2.307927775 + 0.4285714286 = -1.8793563464 v = -1.8793563464 Simplifying v = -1.8793563464

Solution

The solution to the problem is based on the solutions from the subproblems. v = {2.7364992036, -1.8793563464}

Solution

v = {2.7364992036, -1.8793563464}

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