Factoring prob?

Question

Please explain to me how do I factor

7x^2 - 5x - 2 = 0

I need to know how this transforms to (7x-2)(x-1)=0
I know if i foil this out ill get the original. I want to know how do i do it algebriacly instead. Like some steps, thanks

Answer 1

First of all, never use foil. Foil can only be applied to binomials. Once you get to trinomials or anything larger than that, you're in trouble. This is how I multiply two polynomials:

Let's say our terms are A, B, C and D. Our binomials would be (A + B) and (C + D). Ignoring foil, we'll start at the first term of the first binomial and multiply it with each term of the second. We multiply A times C then A times D. We write these out: AC + AD. Then we do the same thing with the next term getting BC + BD. Though this is essentially the same process as foil, you can use it with larger polynomials. A trinomial, for instance, thrown in there might look like this:
(A + B) times (C + D + E)
Foil makes no sense in this case, so we need just use the method above. First A times all the terms in the second: AC + AD + AE, then B times all the terms in the second: BC + BD + BE. Add up AC + AD + AE + BC + BD + BE and you'll get your answer.

Anyway, to answer your actual question: what you need is a trick to figure out which numbers go where. First you must always look for a common factor. In this case there is none, but this must always be the first step. Then, you need recognise that your trinomial has a degree of 2. Thus, each binomial will have a degree of one. This is because each one will have 1 x in it, and multiplying each x together will get you x^2.

So what we now know we have two binomials each with an x, so let's write that down:
( x )( x ) = 0
Now, each x will have a coefficient. Since both x terms multiplied together will produce 7x^2, the coefficients multiplied together must equal 7. Lucky for us, 7 is prime so we know one coefficient must be 7 and the other must be 1, let's write those in:
(7x )(x ) = 0
Now the tricky bit. Each binomial has a constant, multiply these constants together and you get a constant. This is the only way to get a constant in our trinomial above, so each of our constants must multiply together to produce -2. Now, 2 is prime, so we know one constant must be 1 and the other must be 2. Either of these can be negative, so now we need to figure out which, also, in which binomial each belongs.

This is where trial and error comes in. Because of the way we multiply polynomials, -5x must be produced by multiplying a constant by an x term. We have two constants and two x terms, so we will have two products being added together, resulting in -5x. So we know that the negative number must be larger. Since we have 7x in one binomial, anything times that will be the largest term, so the negative must be in the other binomial, let's write this down:
(7x + )(x - ) = 0
Now we use trial and error. We'll put each number 2 and 1 in the two remaining slots and see if they work, let's first try:
(7x + 1)(x - 2) = 0
-2 times 7x gives -14x
1 times x gives x
-14x plus x gives -13x, this is not correct. Let's now try:
(7x + 2)(x - 1) = 0
-1 times 7x gives -7x
2 times x gives 2x
-7x plus 2x gives -5x, this correct. So now we know that:
7x^2 - 5x -2 = 0 can be factored into
(7x + 2)(x - 1) = 0

I hope this helps.

Answer 2

The correct answer has a plus instead of a minus...

7x^2 - 5x - 2 = (7x + 2) (x - 1)

How can you find this answer? First of all a warning: it is not always possible to factor formulas like this. But if there is a factoring, you can find it is follows.

We want to find a, b, u and v in the formula
7x^2 - 5x - 2 = (ax + u) (bx + v)

If you "FOIL" the formula at the right, you get

(ax)(bx) + (ax)v + u(bx) + uv = abx^2 + (av + bu)x + uv

If you compare this with the original formula, it is clear that we must choose the numbers so that

ab = 7
av + bu = -5
uv = -2

Let's start with the first fact: ab = 7. We must find two numbers whose product is 7. There are only two possibilities if a and b are integer numbers: either 1 * 7 or (-1) * (-7).

In the same way, the third fact, uv = -2, has only two possiblities: 1 * (-2) or 2 * (-1).

Now try to combine these conclusions in such a way that the middle fact, av + bu = -5, is also true. So you try

(x + 1) * (7x - 2) = 7x^2 + 5x - 2 ==> almost correct
(x + 2) * (7x - 1) = 7x^2 + 13x - 2 ==> not good
(x - 2) * (7x + 1) = 7x^2 - 13x - 2 ==> not good
(x - 1) * (7x + 2) = 7x^2 - 5x - 2 ==> CORRECT!

Answer 3

what i am going to teach you now is called the sum and product method
multiply the coefficient of the x^2 term and the constant term to get their product.
in this case it will be 7*(-2)=-14
sum will be the coefficient of the x term,
in this case -5
now find all the factors of -14
they will be (1)(-14) or(14)(-1) or(2)*(-7) or (7)(-2).
choose those factors which will add up to the sum that is -5.
in this case it will be -7 and 2.
now split the middle term with coefficients equal to these factors i.e.-7x and 2x
now the given equation can be rewritten as 7x^2-7x+2x-2=0
now taking out the common factors of terms 1&2 and terms3&4
the equation will reduce to 7x(x-1)+2(x-1)=0
and now taking out(x-1) as the common factor
the equation will be (x-1)(7x+2)=0
using the rule that if a product is equal to 0 each and every factor could be zero
we can say either (x-1)=0 =>x=1
or(7x+2)=0giving x=-2/7.
so the solution set is {1,-2/7}
i hope yougot the hang of it

Answer 4

That isn't right....I think it is (7x+2)(x-1). You have to use addition, subtraction, trial and error in order to factor this problem. You multiply first x first, first times second, second times first and second times second. The way you had it, you would have ended up with +2 because -1 x -2 = positive 2. And it has to be 7x+2 rather than 7x-2 because the middle part is negative.

Then after you factor, you can make either side equal to zero and you will have solved for x soooo....

7x+2 = 0
x = -2/7

x - 1 = 0
x = 1

for either x value, the equation will equal 0.

Hope this helps.

Answer 5

3x² + 5x + 2 = 0 the middle time period is + 5x discover the sum of the middle time period Multiply the first time period 3 situations the terrific time period 2 equals 6 and element factors of 6 a million x 6 2 x 3. . . .<=. .Use those factors + 3 and + 2 fulfill the sum of the middle time period Insert + 3x and + 2x into the equation 3x² + 5x + 2 = 0 Gropup element 3x² + 3x + 2x + 2 = 0 3x(x + a million) + 2(x + a million) = 0 (3x + 2)(x + a million) = 0 - - - - - - - - Roots 3x + 2 = 0 3x + 2 - 2 = 0 - 2 3x = - 2 3x / 3 = - 2 / 3 x = - 2/3 - - - - - - - - Roots x + a million = 0 x + a million - a million = 0 - a million x = - a million - - - - - - - - s-