You can run this formula in reverse whenever you are
subtracting two perfect squares . For instance, if we see
, we recognize that both
and 25 are perfect squares. We can therefore factor it as
. Other examples include:
And so on. Note that, in the last example, we begin by
pulling out a 2, and we are then left with two perfect squares. This is an example of the rule that you should always
begin by pulling out common factors before you try anything else!
It is also important to note that you cannot factor the sum of two squares.
is a perfectly good function, but it cannot be factored.
In this case, the multiplication that we are reversing is just FOIL. For instance, consider:
What happened? The 3 and 7
added to yield the middle term (10), and
multiplied to yield the final term
. We can generalize this as:
.
The point is, if you are given a problem such as
to factor, you look for two numbers that
add up to 10, and
multiply to 21. And how do you find them? There are a lot of pairs of numbers that add up to 10, but relatively few that multiply to 21. So you start by looking for factors of 21.
Below is a series of examples. Each example showcases a different aspect of the factoring process, so I would encourage you not to skip over any of them: try each problem yourself, then take a look at what I did.
If you are uncomfortable with factoring, the best practice you can get is to multiply things out . In each case, look at the final answer I arrive at, and multiply it with FOIL. See that you get the problem I started with. Then look back at the steps I took and see how they led me to that answer. The steps will make a lot more sense if you have done the multiplication already.
If the middle term is negative, it doesn’t change much: it just makes both numbers negative. If this had been
, the process would have been the same, and the answer would have been
.
Some things can’t be factored. Many students spend a long time fighting with such problems, but it really doesn’t have to take long. Try all the possibilities, and if none of them works, it can’t be factored.
If the
last term is negative, that changes things! In order to
multiply to –15, the two numbers will have to have different signs—one negative, one positive—which means they will
subtract to give the middle term. Note that if the
middle term were negative, that wouldn’t change the process: the final answer would be reversed,
. This fits the rule that we saw earlier—changing the sign of the middle term changes the answer a bit, but not the process.
Never forget,
always start by looking for common factors to pull out. Then look to see if it fits one of our formulae. Only after trying all that do you begin the FOIL approach.
Factor
What
multiplies to 16?
, or
, or
Which of those adds to 14
after tripling one number ?
If the
has a coefficient, and if you can’t pull it out, the problem is trickier. In this case, we know that the factored form will look like
so we can see that, when we multiply it back, one of those numbers—the one on the right—will be tripled, before they add up to the middle term! So you have to check the number pairs to see if any work that way.
Checking your answers
There are two different ways to check your answer after factoring: multiplying back, and trying numbers.
Problem : Factor
First, pull out the common factor
Difference between two squares
So, does
? First let’s check by multiplying back.
Distributive property
FOIL
Check by trying a number. This should work for any number. I’ll use
and a calculator.
I stress these methods of checking answers, not just because checking answers is a generally good idea, but because they reinforce key concepts. The first method reinforces the idea that
factoring is multiplication done backward . The second method reinforces the idea of algebraic generalizations.
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you.
Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
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