SOLUTION: Find all real or imaginary solutions to each equation. Use the method of your choice. w^2=-225 and 3v^2+4v-1=0

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Question 78127: Find all real or imaginary solutions to each equation. Use the method of your choice.
w^2=-225
and
3v^2+4v-1=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
For each of these problems it's easiest to use the quadratic formula (since there are going to be complex roots):
w%5E2=-225
w%5E2%2B225=0 Add 225 to both sides
Now lets use the quadratic formula:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aw%5E2%2Bbw%2Bc=0 (in our case 1w%5E2%2B0w%2B225+=+0) has the following solutons:

w%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%280%29%5E2-4%2A1%2A225=-900.

The discriminant -900 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.


In the field of imaginary numbers, the square root of -900 is + or - sqrt%28+900%29+=+30.

The solution is w%5B12%5D+=+%28-0%2B-+i%2Asqrt%28+-900+%29%29%2F2%5C1+=++%28-0%2B-+i%2A30%29%2F2%5C1+

Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B0%2Ax%2B225+%29



So we have 2 complex (or imaginary) solutions:
x=15i or x=-15i
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Again lets use the quadratic formula:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation av%5E2%2Bbv%2Bc=0 (in our case 3v%5E2%2B4v%2B-1+=+0) has the following solutons:

v%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%284%29%5E2-4%2A3%2A-1=28.

Discriminant d=28 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-4%2B-sqrt%28+28+%29%29%2F2%5Ca.

v%5B1%5D+=+%28-%284%29%2Bsqrt%28+28+%29%29%2F2%5C3+=+0.21525043702153
v%5B2%5D+=+%28-%284%29-sqrt%28+28+%29%29%2F2%5C3+=+-1.54858377035486

Quadratic expression 3v%5E2%2B4v%2B-1 can be factored:
3v%5E2%2B4v%2B-1+=+3%28v-0.21525043702153%29%2A%28v--1.54858377035486%29
Again, the answer is: 0.21525043702153, -1.54858377035486. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B4%2Ax%2B-1+%29


So we have 2 real solutions:
x=-1.54858377035486 or x=0.21525043702153