9514 1404 393
Answer:
(a) y=(x+4)^2 +7
Step-by-step explanation:
The vertex form equation is ...
y = a(x -h)^2 +k
for vertex (h, k) and vertical scale factor 'a'.
The vertex is given as (h, k) = (-4, 7), so the equation will be ...
y = a(x +4)^2 +7
The value of 'a' can be found from the other given point:
8 = a(-3+4)^2 +7
1 = a
So, the equation is ...
y = (x +4)^2 +7
Answer:
460
Step-by-step explanation:
919 ÷2 = 459.5 so you have to round to the nearest hole so when its 5 or more you round up to the nearest whole and when its 4 or less you round down so your answer is 460
Answer:
-3.52
Step-by-step explanation:
-2.2 x (-2) / (- 1/4) x 5 PEMDAS
4.4 / -1.25
-3.52
Answer:
-ln|x−5| + 2 ln(x²+4) + 3 tan⁻¹(x/2) + C
Step-by-step explanation:
The fraction will be split into a sum of two other fractions.
The first fraction will have a denominator of x − 5. The numerator will the a polynomial of one less order, in this case, a constant A.
The second fraction will have a denominator of x² + 4. The numerator will be Bx + C.
![\frac{3x^{2}-26x+26}{(x-5)(x^{2}+4)}=\frac{A}{x-5} +\frac{Bx+C}{x^{2}+4}](https://tex.z-dn.net/?f=%5Cfrac%7B3x%5E%7B2%7D-26x%2B26%7D%7B%28x-5%29%28x%5E%7B2%7D%2B4%29%7D%3D%5Cfrac%7BA%7D%7Bx-5%7D%20%2B%5Cfrac%7BBx%2BC%7D%7Bx%5E%7B2%7D%2B4%7D)
Combine the two fractions back into one using the common denominator.
![\frac{A}{x-5} +\frac{Bx+C}{x^{2}+4}=\frac{A(x^{2}+4)+(Bx+C)(x-5)}{(x-5)(x^{2}+4)}](https://tex.z-dn.net/?f=%5Cfrac%7BA%7D%7Bx-5%7D%20%2B%5Cfrac%7BBx%2BC%7D%7Bx%5E%7B2%7D%2B4%7D%3D%5Cfrac%7BA%28x%5E%7B2%7D%2B4%29%2B%28Bx%2BC%29%28x-5%29%7D%7B%28x-5%29%28x%5E%7B2%7D%2B4%29%7D)
This numerator will equal the original numerator.
![A(x^{2}+4)+(Bx+C)(x-5)=3x^{2}-26x+26\\Ax^{2}+4A+Bx^{2}-5Bx+Cx-5C=3x^{2}-26x+26\\(A+B)x^{2}+(C-5B)x+(4A-5C)=3x^{2}-26x+26](https://tex.z-dn.net/?f=A%28x%5E%7B2%7D%2B4%29%2B%28Bx%2BC%29%28x-5%29%3D3x%5E%7B2%7D-26x%2B26%5C%5CAx%5E%7B2%7D%2B4A%2BBx%5E%7B2%7D-5Bx%2BCx-5C%3D3x%5E%7B2%7D-26x%2B26%5C%5C%28A%2BB%29x%5E%7B2%7D%2B%28C-5B%29x%2B%284A-5C%29%3D3x%5E%7B2%7D-26x%2B26)
Match the coefficients.
![A+B=3\\C-5B=-26\\4A-5C=26](https://tex.z-dn.net/?f=A%2BB%3D3%5C%5CC-5B%3D-26%5C%5C4A-5C%3D26)
Solve the system of equations.
![A=-1\\B=4\\C=-6](https://tex.z-dn.net/?f=A%3D-1%5C%5CB%3D4%5C%5CC%3D-6)
So we can rewrite the integral as:
![\int {(\frac{-1}{x-5}+\frac{4x-6}{x^{2}+4}) } \, dx](https://tex.z-dn.net/?f=%5Cint%20%7B%28%5Cfrac%7B-1%7D%7Bx-5%7D%2B%5Cfrac%7B4x-6%7D%7Bx%5E%7B2%7D%2B4%7D%29%20%7D%20%5C%2C%20dx)
Solving:
![\int {\frac{-1}{x-5}\, dx + \int {\frac{4x}{x^{2}+4}} \, dx - \int {\frac{6}{x^{2}+4} } \, dx](https://tex.z-dn.net/?f=%5Cint%20%7B%5Cfrac%7B-1%7D%7Bx-5%7D%5C%2C%20dx%20%2B%20%5Cint%20%7B%5Cfrac%7B4x%7D%7Bx%5E%7B2%7D%2B4%7D%7D%20%5C%2C%20dx%20-%20%5Cint%20%7B%5Cfrac%7B6%7D%7Bx%5E%7B2%7D%2B4%7D%20%7D%20%5C%2C%20dx)
![-\int {\frac{1}{x-5}\, dx + 2\int {\frac{2x}{x^{2}+4}} \, dx - 6\int {\frac{1}{x^{2}+4} } \, dx](https://tex.z-dn.net/?f=-%5Cint%20%7B%5Cfrac%7B1%7D%7Bx-5%7D%5C%2C%20dx%20%2B%202%5Cint%20%7B%5Cfrac%7B2x%7D%7Bx%5E%7B2%7D%2B4%7D%7D%20%5C%2C%20dx%20-%206%5Cint%20%7B%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%2B4%7D%20%7D%20%5C%2C%20dx)
![-ln|x-5| + 2ln(x^{2}+4) - 6(\frac{1}{2} tan^{-1}(\frac{x}{2} )) + C](https://tex.z-dn.net/?f=-ln%7Cx-5%7C%20%2B%202ln%28x%5E%7B2%7D%2B4%29%20-%206%28%5Cfrac%7B1%7D%7B2%7D%20tan%5E%7B-1%7D%28%5Cfrac%7Bx%7D%7B2%7D%20%29%29%20%2B%20C)
![-ln|x-5| + 2ln(x^{2}+4) - 3 tan^{-1}(\frac{x}{2} ) + C](https://tex.z-dn.net/?f=-ln%7Cx-5%7C%20%2B%202ln%28x%5E%7B2%7D%2B4%29%20-%203%20tan%5E%7B-1%7D%28%5Cfrac%7Bx%7D%7B2%7D%20%29%20%2B%20C)