We have the expressions:
![f(x)=x^2+6x+15](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E2%2B6x%2B15)
![y=x^2-4x+9](https://tex.z-dn.net/?f=y%3Dx%5E2-4x%2B9)
![f(x)=x^2-8x](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E2-8x)
![y=x^2+7x-2](https://tex.z-dn.net/?f=y%3Dx%5E2%2B7x-2)
Now, with this we operate as follows:
a)
![f(x)=x^2+6x+15=x^2+6x+9+15-9](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E2%2B6x%2B15%3Dx%5E2%2B6x%2B9%2B15-9)
![\Rightarrow(x+3)^2+6](https://tex.z-dn.net/?f=%5CRightarrow%28x%2B3%29%5E2%2B6)
Then, the axis is x = -3 and the vertex (-3, 6)
b)
![y=x^2-4x+9=x^2-4x+4+9-4](https://tex.z-dn.net/?f=y%3Dx%5E2-4x%2B9%3Dx%5E2-4x%2B4%2B9-4)
![\Rightarrow(x-2)^2+5](https://tex.z-dn.net/?f=%5CRightarrow%28x-2%29%5E2%2B5)
Then, the vertex is (2, 5) and the axis is x = 2.
c)
![f(x)=x^2-8x+4-4\Rightarrow(x-2)^2-4](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E2-8x%2B4-4%5CRightarrow%28x-2%29%5E2-4)
Then, the vertex is (2, -4) andd the axis is x = 2.
d)
![y=x^2+7x-2=x^2+7x+\frac{49}{4}-2-\frac{49}{4}](https://tex.z-dn.net/?f=y%3Dx%5E2%2B7x-2%3Dx%5E2%2B7x%2B%5Cfrac%7B49%7D%7B4%7D-2-%5Cfrac%7B49%7D%7B4%7D)
![\Rightarrow(x+\frac{7}{2})^2-\frac{57}{4}](https://tex.z-dn.net/?f=%5CRightarrow%28x%2B%5Cfrac%7B7%7D%7B2%7D%29%5E2-%5Cfrac%7B57%7D%7B4%7D)
Then, the vertex is (-7/2, -57/4) and the axis is -7/2.
Answer:
the second one
Step-by-step explanation:
2 halves make a whole. 1/2 + 1/2
4 fourths make a whole. 1/4 + 1/4 + 1/4 + 1/4
a half of a fourth is an eighth. 1/4 ÷ 1/2 = 1/8
Answer:
Lola has 8 slices.
Step-by-step explanation:
Answer:
![e^x+xy+3y+(y-1)e^y=4](https://tex.z-dn.net/?f=e%5Ex%2Bxy%2B3y%2B%28y-1%29e%5Ey%3D4)
Step-by-step explanation:
Given that
![(e^x+y)dx+(3+x+ye^y)dy=0](https://tex.z-dn.net/?f=%28e%5Ex%2By%29dx%2B%283%2Bx%2Bye%5Ey%29dy%3D0)
Here
![M=e^x+y](https://tex.z-dn.net/?f=M%3De%5Ex%2By)
![N=3+x+ye^y](https://tex.z-dn.net/?f=N%3D3%2Bx%2Bye%5Ey)
We know that
M dx + N dy=0 will be exact if
![\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20M%7D%7B%5Cpartial%20y%7D%3D%5Cfrac%7B%5Cpartial%20N%7D%7B%5Cpartial%20x%7D)
So
![\frac{\partial M}{\partial y}=1](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20M%7D%7B%5Cpartial%20y%7D%3D1)
![\frac{\partial N}{\partial x}=1](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20N%7D%7B%5Cpartial%20x%7D%3D1)
it means that this is a exact equation.
![\int d\left(e^x+xy+3y+(y-1)e^y\right)=0](https://tex.z-dn.net/?f=%5Cint%20d%5Cleft%28e%5Ex%2Bxy%2B3y%2B%28y-1%29e%5Ey%5Cright%29%3D0)
Noe by integrating above equation
![e^x+xy+3y+(y-1)e^y=C](https://tex.z-dn.net/?f=e%5Ex%2Bxy%2B3y%2B%28y-1%29e%5Ey%3DC)
Given that
x= 0 then y= 1
![e^0+0+3+(1-1)e^1=C](https://tex.z-dn.net/?f=e%5E0%2B0%2B3%2B%281-1%29e%5E1%3DC)
C=4
So the our final equation will be
![e^x+xy+3y+(y-1)e^y=4](https://tex.z-dn.net/?f=e%5Ex%2Bxy%2B3y%2B%28y-1%29e%5Ey%3D4)