SOLUTION: 1. x2 - 8x + 16 = 36 2. use the discriminant to determine whether the following equations have solutions that are: two different rational solutions; two different irrational solut

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: 1. x2 - 8x + 16 = 36 2. use the discriminant to determine whether the following equations have solutions that are: two different rational solutions; two different irrational solut      Log On


   

Question 248831: 1. x2 - 8x + 16 = 36
2. use the discriminant to determine whether the following equations have solutions that are: two different rational solutions; two different irrational solutions; exactly one rational solution; or two different imaginary solutions. 3 + 8z2 = -7z
Answer by oberobic(2304) About Me  (Show Source):
You can put this solution on YOUR website!
1.
x%5E2+-+8x+%2B+16+=+36
subtract 36 from both sides
x%5E2+-8x+-20+=+0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-8x%2B-20+=+0) has the following solutons:

x%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=%28-8%29%5E2-4%2A1%2A-20=144.

Discriminant d=144 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--8%2B-sqrt%28+144+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-8%29%2Bsqrt%28+144+%29%29%2F2%5C1+=+10
x%5B2%5D+=+%28-%28-8%29-sqrt%28+144+%29%29%2F2%5C1+=+-2

Quadratic expression 1x%5E2%2B-8x%2B-20 can be factored:
1x%5E2%2B-8x%2B-20+=+1%28x-10%29%2A%28x--2%29
Again, the answer is: 10, -2. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-8%2Ax%2B-20+%29

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2.
3+%2B+8z%5E2+=+-7z
Add 7z to both sides and rearrange
8z%5E2+%2B+7z+%2B+3+=+0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation az%5E2%2Bbz%2Bc=0 (in our case 8z%5E2%2B7z%2B3+=+0) has the following solutons:

z%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=%287%29%5E2-4%2A8%2A3=-47.

The discriminant -47 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 -47 is + or - sqrt%28+47%29+=+6.85565460040104.

The solution is

Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+8%2Ax%5E2%2B7%2Ax%2B3+%29