Please Help: Understanding Quadratic Equations for Factoring?

Question

I don't understand where the middle number comes from, the number times x to the first power. It looks like you're doing addition when you factor the problem, but it seems like it should be only multiplication.

Example: 2x squared + 3x + 6 = 0. Where does that 3x come from or go to, because when you factor it out, you wind up with an answer like (x + something)(x + something)? The two x's times each other makes x squared. The middle number (3x in example)...?

Thanks for the help. I'm trying to learn this (again), but I'm stuck on understanding this point.

Samuel D, how do you get a squared sign using your keyboard? I'd love to know!

Sahsjing and Tess (and maybe Kalkmat? not sure), thank you for being sharp enough to catch that apparently the equation I made up had a problem with it. I just pulled numbers out of the air as an example; I guess I didn't pick so well! Thank you for pointing that out, to spare me further confusion in the future.

Answer 1

( I just made this problem up:)
For x^2-2x-24, the factored version is this:

(x-6)(x+4)

The -2x in the middle comes from FOILing
F- First
O-outside
I-inside
L-last

That's how you multiply the factors.
When you multiply the first in each factor you get:
x^2
Multiply the outsides you get:
+4x
Multiply the insides:
-6x
Multiply the lasts:
-24

So join that together:
x^2 +4x-6x -24
You can join the +4x and -6x which equals ==> -2x
So when you multiply it out, and group the same terms, you get :
x^2-2x-24

Do you get it?
If you need any extra examples or websites or anything, just email me :D

Answer 2

A quadratic equation is in the form ax^ + bx + c = 0 and this tells us that there are 2 x values that is the graph touches or intersects the x-axis at two points as it is a parabolla. If you are factoring out, you have to:

1. Find the product of ac
2. Find the two numbers that give a product of ac and a sum of b
3. Write b as a sum of the two numbers in 2
4. Factor in pairs

Example: 2x^2 + 7x +6 = 0

ac = 2 * 6 = 12
two numbers which give a product of 12 and a sum of 7 are 3 and 4

2x^2 + 7x + 6 = 0
2x^2 +4x + 3x + 6 = 0
2x(x + 2) + 3(x + 2) = 0
(2x + 3)(x + 2) = 0
2x + 3 = 0 or x + 2 = 0
x = -3/2 or x = -2

2x^2 + 3x + 6 can not be factorised, hence to solve that equation you can use a formula or solve by completing the square

Answer 3

Factoring

x² + 5x + 6 = 0

The middle term is + 5x

Find the sum of the middle term

The first term x² = 1

Multiply the first terms 1 times the last term 6 equals 6 and factor

The factors of 6

1 x 6
2 + 3. . .←. .use these factors

+ 2 and + 3 satisfy the sum of the middle term

Insert + 2x and + 3x into the equation

x² + 5x + 6 = 0

x² + 2x + 3x + 6 = 0

Group factor

(x² + 2x) + (3x + 6) = 0

x(x + 2) + 3(x + 2) = 0

(x + 3)(x + 2) =

- - - - - -

Roots

x + 2 = 0

x + 2 - 2 = 0 - 2

x = - 2

- - - - --

Roots

x + 3 = 0

x + 3 - 3 = 0 - 3

x = - 3

- - - - - -s-

Answer 4

Let assume you have the equation x^2 + 5x + 6. The factors are (x+3) and (x+2). When you multiply the 2 factors together, both parts of the first have to multiply with both parts of the second, not just first times first and second times second. Try to remember the term FOIL = FIRST OUTSIDE INSIDE LAST.

(x+2)(x+3)

FIRST = (x)(x) = x^2
OUTSIDE = (x)(3) = 3x
INSIDE = (2)(x) = 2x
LAST = (2)(3) = 6

Then you add the like terms together (3x + 2x = 5x) so your final equation is:

x^2 + 5x + 6.

Hope this helped.

Answer 5

2x^2+3x+6 is unfactorable in real domain because b^2-4ac < 0.

But you can factor 2x^2+7x+6.
2x^2+7x+6 = (2x+3)(x+2)
-----------
Ideas:
2(6) = 12, the product of the leading coefficient and constant term.
12 = 3*4
3+4 = 7, the coefficient of the linear term, which means 7x comes from the sum of 3x and 4x.

Answer 6

you need to bear in mind that a quadratic equation need to obey certain rules before it can be factorised into (x+y)(x+z) form. yup the b^2 - 4ac need to be above 0. now if u asked where does the ''number times x to the first power'' comes from, try expanding (ax+y)(bx+z)

ill do it for you:

axbx +ybx +axz +yz
becomes: (ab)x^2 +(yb+az)x +yz

now u see that the ''number times x to the first power'' is actually number as a result of combination of functions (multiplication and addition). well actually it has more complex derivation for a more complex equation but for the sake of understanding it.. i think this would do just fine. hope this helps.

Answer 7

The method you use is FOIL. First, outer, inner, and last. When you multiply the the "firsts," they become x squared. However, the outer and the inner will become in this example (+somethingx and +something x). When you add these two together, you get (+2 something x) because when you add variables of the same power, they stay at that same power. Hope this helps.