SOLUTION: Find all real or imaginary solutions to each equation. Use the method of your choice. 3v^2 + 4v - 1 = 0

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Find all real or imaginary solutions to each equation. Use the method of your choice. 3v^2 + 4v - 1 = 0      Log On


   

Question 98540: Find all real or imaginary solutions to each equation. Use the method of your choice.
3v^2 + 4v - 1 = 0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's use the quadratic formula to solve for v:


Starting with the general quadratic

av%5E2%2Bbv%2Bc=0

the general solution using the quadratic equation is:

v+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve 3%2Av%5E2%2B4%2Av-1=0 ( notice a=3, b=4, and c=-1)

v+=+%28-4+%2B-+sqrt%28+%284%29%5E2-4%2A3%2A-1+%29%29%2F%282%2A3%29 Plug in a=3, b=4, and c=-1



v+=+%28-4+%2B-+sqrt%28+16-4%2A3%2A-1+%29%29%2F%282%2A3%29 Square 4 to get 16



v+=+%28-4+%2B-+sqrt%28+16%2B12+%29%29%2F%282%2A3%29 Multiply -4%2A-1%2A3 to get 12



v+=+%28-4+%2B-+sqrt%28+28+%29%29%2F%282%2A3%29 Combine like terms in the radicand (everything under the square root)



v+=+%28-4+%2B-+2%2Asqrt%287%29%29%2F%282%2A3%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



v+=+%28-4+%2B-+2%2Asqrt%287%29%29%2F6 Multiply 2 and 3 to get 6

So now the expression breaks down into two parts

v+=+%28-4+%2B+2%2Asqrt%287%29%29%2F6 or v+=+%28-4+-+2%2Asqrt%287%29%29%2F6


Now break up the fraction


v=-4%2F6%2B2%2Asqrt%287%29%2F6 or v=-4%2F6-2%2Asqrt%287%29%2F6


Simplify


v=-2+%2F+3%2Bsqrt%287%29%2F3 or v=-2+%2F+3-sqrt%287%29%2F3


So these expressions approximate to

v=0.21525043702153 or v=-1.54858377035486


So our solutions are:
v=0.21525043702153 or v=-1.54858377035486

Notice when we graph 3%2Ax%5E2%2B4%2Ax-1 (just replace v with x), we get:



when we use the root finder feature on a calculator, we find that x=0.21525043702153 and x=-1.54858377035486.So this verifies our answer