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Usually, the easiest way to do this is to find a pair of numbers whose sum is equal to the coefficient of the first term, and whose product is equal to the product of the coefficients of the first and last terms:
a + b = -3
a * b = -10
take all possible pairs of numbers that multiply to make negative ten:
1 * -10 = -10
-1 * 10 = -10
2 * -5 = -10
-2 * 5 = -10
and then find which one of those pairs also adds up to negative three:
1 + (-10) = -9
-1 + 10 = 9
2 + (-5) = -3
-2 + 5 = 3
So 2 and -5 are the coefficients we want. Now take our original equation:
10a2 - 3a - 1
and separate the middle term into two terms using those coefficients:
10a2 - 5a + 2a - 1
You can then factor a common term out of the first pair and last pair of terms:
5a(2a - 1) + 1(2a - 1)
and then group your coefficients:
(5a + 1)(2a - 1)
Giving you the answer.
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SteveWe have the algebraic expression: 8a2 - 2a
In order to factorize it, we can just simply pull out 2a and we are now left with this: 2a(4a-1)
But, why 2a?
In the given algebraic expression we have two terms: 8a2 and -2a
Now, we have to look for the common factor of 8a2 and -2a.
8a2 = 8 x a x a = 2 x 2 x 2 x a x a
-2a = -2 x a
Common factor is 2a. (2 is also the common factor but we need greatest common factor which is 2a)
4a - 1 can not be further factorized so the final answer is 2a(4a-1).
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What is the factor tree and prime factorization fo the number 54?
54 27,2 9,3,2 3,3,3,2 2 x 3 x 3 x 3 = 54
What is the gcf of 8 and 24 and 20?
The Greatest Common Factor fo 20 and 24 is 4. This is because 4x5=20 and 4x6=24.
What is the factor of two numbers?
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What is the gcf fo 64 and 72?
It is: 8
How does a tank circuit filter out frequencies?
Without writing a book, the tank circuit (tank) is what is called a tuned circuit. The basic tank circuit has an inductive component (a coil) and a capacitive component (a cap). There are a couple of concepts that you need to get a handle on. A tank has a given "resonance" based on the net inductance and capacitance. It has one frequency that it really likes to run at. That's the resonant frequency (fo). The tank is tuned for that fo. A signal at that frequency that is put into the stage that has a tank filter will cause the tank to "ring" electronically. The signal oscillates in the tank, and, because it's the fo for that tank, some maximum amplitude (gain) will be attained. The signal is then coupled out. Other signals of a slightly different frequency will cause the tank to ring, but not as much. As a signal is put into the stage that is a bit higher or lower than the fo, it causes less oscillation, and it emerges at a lower amplitude. Notice that this applies to signals that are a bit above or a bit lower than the fo. The farther from the fo that a given signal is, the less oscillation it produces in the tank, and the less gain it will be given in that stage. Again (and it's worth repeating), signals both above and below the fo a given distance produce the same oscillation in the tank. That means the tank responds to both sides of the spectrum above and below the fo. Lastly, the tank can be "sharply tuned" or not. In a sharply tuned tank, the signals that are farther from fo cause little tank action, and those right around fo really get a lot of gain. In something less sharpely tuned, there is still a good bit of gain for frequencies higher or lower than the fo. We spoke of the tuning of the tank for a particular fo. Now we have to tune the tank for what is called quality (Q). This speaks to how "tightly" the tank is tuned. And we have to make a trade. If we want a lot gain at fo and little else, high Q is the way to go. There will be a ton of gain at fo, and the gain for frequencies above and below will drop like a rock. In a less tightly tuned tank, there will be better gain for the frequencies around fo, but not as great a gain for the fo itself as there would be in a high Q tank. That's the trade. Narrow tuning yields greater gain for the fo and lower "bandpass" while broader tuning will pass "more frequencies" (have a higher band pass) but with lower gain at and right around the fo than a high Q tank.
