Problem involving factoring?

Question

Factor : a^2 + 2a – 3

Can you explain factoring, especially how you know what signs to put in each part. Such as: x^2 - 5x + 6 is factored to (x - 2)(x - 3). How do you figure out that both with be subtraction? Thanks.

*will (not with)

Answer 1

If you have x^2 -5x + 6, to factor you want to find two numbers such that those two numbers multiply together to give you 6 AND add together to give -5.

More generally, when you have x^2 + ax + b, you want two numbers, c and d, that multiply to give you b and the same two numbers add to give you a. Then x^2 + ax + b = (a+c)(b+d) Make sense?

So what two numbers multiply together to give you 6 and add together to give you -5? Well a negative times a negative is a positive, so let's try -2 and -3.

-2 times -3 = 6. -2 + -3 = -5. And those are your two factors!

Now, a^2 + 2a - 3. What two numbers multiply together to give you -3 and add together to give you 2?

Well, 3 times -1 gives you negative 3. So far so good. Does 3 + -1 give you 2? 3 + -1 is the same as 3 -1 = 2. so that's it!
a^2 + 2a - 3 = (a + 3)(a - 1)

Answer 2

The best thing about factoring is that it's easy to check. After you do it and as you want to put the signs in check to see that it works when you multiply it back.

Let's look at a^2 + 2a -3.

I'm looking for two numbers that multiply to give me -3, which means their signs must be different. It's either 3 and -1 or -3 and +1. But they must add to give me that +2 that's multiplying a. Okay, then it has to be +3 and -1:

(a + 3)(a - 1) and it's done.

As for your second example, x^2 - 5x + 6, see how they have to add to -5 so they both have to be negative.

The more you do the easier it gets because you're teaching your brain to just do them. At first it's so hard, I know, but I promise you it will get easier.

Answer 3

(a+3)(a-1)

In your equation above x^2-5x+6

They are both subtraction in this equation because of the factors necessary to complete the equation correctly.
A negative times a negative = a positive

Since you have a negative before the second sign and a positive for the 3rd sign it is logical to assumme you will need to multiply a negative by a negative to get thr 3rd figure to be a positive

X*x=x^2
-2*x=-2x
-3*x=-3x
-3*-2=6
X^2-5x+6

Answer 4

a² + 2a - 3 = 0

The middle term is + 2a

Find the sum of the middoe term

The first term a² = 1

Multiply the first term 1 times the last term 3 Equals 3 and factor

Factors of 3

1 x 3. . .←. .use these factors

+ 3 and - 1 satisfy the sum of the middle term

insert + 3a and - 1a into the equation

a² + 2a - 3 = 0

a² + 3a - 1a - 3 = 0

Group factor

(a² + 3a) - (1a - 3 = 0

a(a + 3) - 1(a + 3) = 0

(a - 1)(a + 3) = 0

- - - - - - - - s-

Answer 5

If the sign at the end is positive (like your +6), then you know that both factors will be the same...either ++ or --. That's because you're looking for 2 numbers that multiply to +6. The only way to multiply to a positive is positive times positive or negative times negative.

So, you know they're the same sign, but they also have to add up to the middle term, which is -5. If the signs have to add up to a negative, and they're the same sign, it has to be -3, -2.

For your example with the "a" variables. The sign at the end is negative. You're looking for 2 numbers that multiply to -3. In order to end up with a negative from multiplying, one number must be positive, one number must be negative. Also, since they're going to add up to a +2 (since that's your middle term), the bigger number must be posiitve.
(a+3)(a-1). Those numbers multiply to -3, combine (add to) +2.

Answer 6

a^2 + 2a – 3 = (a+3)(a-1)
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Ideas: Factor -3 into 3 and -1 since 3-1 = 2, the coefficient of "a".