The circumference is the correct.
It can be found using the formula,
Answer: It has two distinct real zeros.
Step-by-step explanation:
The formula that is used to calculate the discriminant of a Quadratic function is the one shown below:
![D=b^2-4ac](https://tex.z-dn.net/?f=D%3Db%5E2-4ac)
In this case you have the following Quadractic function provided in the exercise:
![f(x)=6x^2 +10x-1](https://tex.z-dn.net/?f=f%28x%29%3D6x%5E2%20%2B10x-1)
Let's make it equal to 0:
![0=6x^2 +10x-1](https://tex.z-dn.net/?f=0%3D6x%5E2%20%2B10x-1)
You can identify that:
![a=6\\\\b=10](https://tex.z-dn.net/?f=a%3D6%5C%5C%5C%5Cb%3D10)
Knowing these values, you can substitute them into the formula and then evaluate:
![D=10^2-4(6)(-1)\\\\D=124](https://tex.z-dn.net/?f=D%3D10%5E2-4%286%29%28-1%29%5C%5C%5C%5CD%3D124)
Therefore, since:
![D>0](https://tex.z-dn.net/?f=D%3E0)
You can determine that the it has two distinct real roots.
To simplify
![\sqrt[4]{\dfrac{24x^6y}{128x^4y^5}}](https://tex.z-dn.net/?f=%5Csqrt%5B4%5D%7B%5Cdfrac%7B24x%5E6y%7D%7B128x%5E4y%5E5%7D%7D)
we need to use the fact that
![\sqrt[4]{x^4}=|x|](https://tex.z-dn.net/?f=%5Csqrt%5B4%5D%7Bx%5E4%7D%3D%7Cx%7C)
Why the absolute value? It's because
.
We start by rewriting as
![\sqrt[4]{\dfrac{2^23x^6y}{2^6x^4y^5}}](https://tex.z-dn.net/?f=%5Csqrt%5B4%5D%7B%5Cdfrac%7B2%5E23x%5E6y%7D%7B2%5E6x%5E4y%5E5%7D%7D)
![\sqrt[4]{\dfrac{2^23x^4x^2y}{2^42^2x^4y^4y}}](https://tex.z-dn.net/?f=%5Csqrt%5B4%5D%7B%5Cdfrac%7B2%5E23x%5E4x%5E2y%7D%7B2%5E42%5E2x%5E4y%5E4y%7D%7D)
Since
, we have
, and the above reduces to
![\sqrt[4]{\dfrac{3x^2y}{2^4y^4y}}](https://tex.z-dn.net/?f=%5Csqrt%5B4%5D%7B%5Cdfrac%7B3x%5E2y%7D%7B2%5E4y%5E4y%7D%7D)
Then we pull out any 4th powers under the radical, and simplify everything we can:
![\dfrac1{\sqrt[4]{2^4y^4}}\sqrt[4]{\dfrac{3x^2y}{y}}](https://tex.z-dn.net/?f=%5Cdfrac1%7B%5Csqrt%5B4%5D%7B2%5E4y%5E4%7D%7D%5Csqrt%5B4%5D%7B%5Cdfrac%7B3x%5E2y%7D%7By%7D%7D)
![\dfrac1{|2y|}\sqrt[4]{3x^2}](https://tex.z-dn.net/?f=%5Cdfrac1%7B%7C2y%7C%7D%5Csqrt%5B4%5D%7B3x%5E2%7D)
where
allows us to write
, and this also means that
. So we end up with
![\dfrac{\sqrt[4]{3x^2}}{2y}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Csqrt%5B4%5D%7B3x%5E2%7D%7D%7B2y%7D)
making the last option the correct answer.
The formula for the area of a triangle is the base times the height, divided by 2. Or in simple forms, bh/2