SOLUTION: Help I am so confused. 1) 2x^2=6 what are the solutions x= what are the x intercepts 2) 10T^2-8T = 0 What is the nature of the solution 3) what is the ve

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Question 194985: Help I am so confused.
1) 2x^2=6 what are the solutions x=
what are the x intercepts
2) 10T^2-8T = 0 What is the nature of the solution
3) what is the vertex of f(x)=-x^2+12x+3

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Help I am so confused.
1) 2x^2=6 what are the solutions x=
what are the x intercepts
The x-intercepts are the solutions.
2x^2 = 6
x^2 = 3
x = ± sqrt(3)
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B0x%2B-3+=+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=%280%29%5E2-4%2A1%2A-3=12.

Discriminant d=12 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-0%2B-sqrt%28+12+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%280%29%2Bsqrt%28+12+%29%29%2F2%5C1+=+1.73205080756888
x%5B2%5D+=+%28-%280%29-sqrt%28+12+%29%29%2F2%5C1+=+-1.73205080756888

Quadratic expression 1x%5E2%2B0x%2B-3 can be factored:
1x%5E2%2B0x%2B-3+=+%28x-1.73205080756888%29%2A%28x--1.73205080756888%29
Again, the answer is: 1.73205080756888, -1.73205080756888. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B0%2Ax%2B-3+%29

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Ignore the factoring statement.
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2) 10T^2-8T = 0 What is the nature of the solution
It's a parabola with the vertex (maximum) at the top (assuming T is horizontal).
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3) what is the vertex of f(x)=-x^2+12x+3
The vertex is at x = -b/2a
or x = -12/(12) = 6
At x = 6, f(x) = -36 + 72 + 3 = 39
--> (-6,39)