Answer:
![5 {x}^{2} - x + 6](https://tex.z-dn.net/?f=5%20%7Bx%7D%5E%7B2%7D%20%20-%20x%20%2B%206)
Step-by-step explanation:
The solution is in the image
If two chords intersect each other inside a circle, the products of their segments are equal.
If M is midpoint of AB, then AM = MB = 24/2 = 12.
<span>Product of the lengths of the segments AM and MB:
</span>AM * MB = 12 * 12 = 144
<span>So, any chord of circle O that intersects AB at its midpoint, M, wll be separated by M into two segments such that the product of the lengths of the segments is 144.
</span>
Answer:
the first one
Step-by-step explanation:
Answer:
![y=3e^{-4t}](https://tex.z-dn.net/?f=y%3D3e%5E%7B-4t%7D)
Step-by-step explanation:
![y''+5y'+4y=0](https://tex.z-dn.net/?f=y%27%27%2B5y%27%2B4y%3D0)
Applying the Laplace transform:
![\mathcal{L}[y'']+5\mathcal{L}[y']+4\mathcal{L}[y']=0](https://tex.z-dn.net/?f=%5Cmathcal%7BL%7D%5By%27%27%5D%2B5%5Cmathcal%7BL%7D%5By%27%5D%2B4%5Cmathcal%7BL%7D%5By%27%5D%3D0)
With the formulas:
![\mathcal{L}[y'']=s^2\mathcal{L}[y]-y(0)s-y'(0)](https://tex.z-dn.net/?f=%5Cmathcal%7BL%7D%5By%27%27%5D%3Ds%5E2%5Cmathcal%7BL%7D%5By%5D-y%280%29s-y%27%280%29)
![\mathcal{L}[y']=s\mathcal{L}[y]-y(0)](https://tex.z-dn.net/?f=%5Cmathcal%7BL%7D%5By%27%5D%3Ds%5Cmathcal%7BL%7D%5By%5D-y%280%29)
![\mathcal{L}[x]=L](https://tex.z-dn.net/?f=%5Cmathcal%7BL%7D%5Bx%5D%3DL)
![s^2L-3s+5sL-3+4L=0](https://tex.z-dn.net/?f=s%5E2L-3s%2B5sL-3%2B4L%3D0)
Solving for ![L](https://tex.z-dn.net/?f=L)
![L(s^2+5s+4)=3s+3](https://tex.z-dn.net/?f=L%28s%5E2%2B5s%2B4%29%3D3s%2B3)
![L=\frac{3s+3}{s^2+5s+4}](https://tex.z-dn.net/?f=L%3D%5Cfrac%7B3s%2B3%7D%7Bs%5E2%2B5s%2B4%7D)
![L=\frac{3(s+1)}{(s+1)(s+4)}](https://tex.z-dn.net/?f=L%3D%5Cfrac%7B3%28s%2B1%29%7D%7B%28s%2B1%29%28s%2B4%29%7D)
![L=\frac3{s+4}](https://tex.z-dn.net/?f=L%3D%5Cfrac3%7Bs%2B4%7D)
Apply the inverse Laplace transform with this formula:
![\mathcal{L}^{-1}[\frac1{s-a}]=e^{at}](https://tex.z-dn.net/?f=%5Cmathcal%7BL%7D%5E%7B-1%7D%5B%5Cfrac1%7Bs-a%7D%5D%3De%5E%7Bat%7D)
![y=3\mathcal{L}^{-1}[\frac1{s+4}]=3e^{-4t}](https://tex.z-dn.net/?f=y%3D3%5Cmathcal%7BL%7D%5E%7B-1%7D%5B%5Cfrac1%7Bs%2B4%7D%5D%3D3e%5E%7B-4t%7D)
Answer: 1. b=-3/2 or -1 1/2
2. B=17/3 or 5 2/3
3. K=-14
Step-by-step explanation: