Call x is the number of girls in the classroom.
12/120 = 25/x
x = 120*25/12
x=250
Answer 250 girsl
Let’s make an equation
Let’s say x = amount of years after
Smith = 8, mother = 42
3(8 + x) = 42 + x
24 + 3x = 42 + x
2x = 18, x = 9
Solution: after 9 years
Answer:
Points apply to the given situation:
(a)The puppy will run into the wall after 5 seconds
(b)The slope is -4 ft per second since the puppy is running at a rate of 4 ft per second.
(c)The y-intercept is 20 ft because the puppy is 20 feet away from the wall.
Step-by-step explanation:
We have given,
A puppy is running at a rate = 4 feet per second
A wall is 20 feet away from puppy. That means initially puppy is 20 feet away from the wall.
So, time taken by puppy to reach the wall =
i.e. time take by puppy to reach the wall = = 5 seconds
Now we write the points that apply to this situation:
(a)The puppy will run into the wall after 5 seconds
(b)The slope is -4 ft per second since the puppy is running at a rate of 4 ft per second. {Since the puppy is moving towards the wall that means horizontal distance is decreasing at a rate of -4 feet per second}
(c)The y-intercept is 20 ft because the puppy is 20 feet away from the wall.
The answer is 130 degrees
![\dfrac{5x^4-7x^3-12x^2+6x+21}{(x-3)(x^2-2)^2}=\dfrac{a_1}{x-3}+\dfrac{a_2x+a_3}{x^2-2}+\dfrac{a_4x+a_5}{(x^2-2)^2}](https://tex.z-dn.net/?f=%5Cdfrac%7B5x%5E4-7x%5E3-12x%5E2%2B6x%2B21%7D%7B%28x-3%29%28x%5E2-2%29%5E2%7D%3D%5Cdfrac%7Ba_1%7D%7Bx-3%7D%2B%5Cdfrac%7Ba_2x%2Ba_3%7D%7Bx%5E2-2%7D%2B%5Cdfrac%7Ba_4x%2Ba_5%7D%7B%28x%5E2-2%29%5E2%7D)
![\implies 5x^4-7x^3-12x^2+6x+21=a_1(x^2-2)^2+(a_2x+a_3)(x-3)(x^2-2)+(a_4x+a_5)(x-3)](https://tex.z-dn.net/?f=%5Cimplies%205x%5E4-7x%5E3-12x%5E2%2B6x%2B21%3Da_1%28x%5E2-2%29%5E2%2B%28a_2x%2Ba_3%29%28x-3%29%28x%5E2-2%29%2B%28a_4x%2Ba_5%29%28x-3%29)
When
![x=3](https://tex.z-dn.net/?f=x%3D3)
, you're left with
![147=49a_1\implies a_1=\dfrac{147}{49}=3](https://tex.z-dn.net/?f=147%3D49a_1%5Cimplies%20a_1%3D%5Cdfrac%7B147%7D%7B49%7D%3D3)
When
![x=\sqrt2](https://tex.z-dn.net/?f=x%3D%5Csqrt2)
or
![x=-\sqrt2](https://tex.z-dn.net/?f=x%3D-%5Csqrt2)
, you're left with
![\begin{cases}17-8\sqrt2=(\sqrt2a_4+a_5)(\sqrt2-3)&\text{for }x=\sqrt2\\17+8\sqrt2=(-\sqrt2a_4+a_5)(-\sqrt2-3)\end{cases}\implies\begin{cases}-5+\sqrt2=\sqrt2a_4+a_5\\-5-\sqrt2=-\sqrt2a_4+a_5\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D17-8%5Csqrt2%3D%28%5Csqrt2a_4%2Ba_5%29%28%5Csqrt2-3%29%26%5Ctext%7Bfor%20%7Dx%3D%5Csqrt2%5C%5C17%2B8%5Csqrt2%3D%28-%5Csqrt2a_4%2Ba_5%29%28-%5Csqrt2-3%29%5Cend%7Bcases%7D%5Cimplies%5Cbegin%7Bcases%7D-5%2B%5Csqrt2%3D%5Csqrt2a_4%2Ba_5%5C%5C-5-%5Csqrt2%3D-%5Csqrt2a_4%2Ba_5%5Cend%7Bcases%7D)
Adding the two equations together gives
![-10=2a_5](https://tex.z-dn.net/?f=-10%3D2a_5)
, or
![a_5=-5](https://tex.z-dn.net/?f=a_5%3D-5)
. Subtracting them gives
![2\sqrt2=2\sqrt2a_4](https://tex.z-dn.net/?f=2%5Csqrt2%3D2%5Csqrt2a_4)
,
![a_4=1](https://tex.z-dn.net/?f=a_4%3D1)
.
Now, you have
![5x^4-7x^3-12x^2+6x+21=3(x^2-2)^2+(a_2x+a_3)(x-3)(x^2-2)+(x-5)(x-3)](https://tex.z-dn.net/?f=5x%5E4-7x%5E3-12x%5E2%2B6x%2B21%3D3%28x%5E2-2%29%5E2%2B%28a_2x%2Ba_3%29%28x-3%29%28x%5E2-2%29%2B%28x-5%29%28x-3%29)
![5x^4-7x^3-12x^2+6x+21=3x^4-11x^2-8x+27+(a_2x+a_3)(x-3)(x^2-2)](https://tex.z-dn.net/?f=5x%5E4-7x%5E3-12x%5E2%2B6x%2B21%3D3x%5E4-11x%5E2-8x%2B27%2B%28a_2x%2Ba_3%29%28x-3%29%28x%5E2-2%29)
![2x^4-7x^3-x^2+14x-6=(a_2x+a_3)(x-3)(x^2-2)](https://tex.z-dn.net/?f=2x%5E4-7x%5E3-x%5E2%2B14x-6%3D%28a_2x%2Ba_3%29%28x-3%29%28x%5E2-2%29)
By just examining the leading and lagging (first and last) terms that would be obtained by expanding the right side, and matching these with the terms on the left side, you would see that
![a_2x^4=2x^4](https://tex.z-dn.net/?f=a_2x%5E4%3D2x%5E4)
and
![a_3(-3)(-2)=6a_3=-6](https://tex.z-dn.net/?f=a_3%28-3%29%28-2%29%3D6a_3%3D-6)
. These alone tell you that you must have
![a_2=2](https://tex.z-dn.net/?f=a_2%3D2)
and
![a_3=-1](https://tex.z-dn.net/?f=a_3%3D-1)
.
So the partial fraction decomposition is