How do you factor x^2-25x+100?

Question

Show work to help me out?

Answer 1

(x-20)(x-5)

You may have already done some problems like this:

(x - 20)(x - 5)

Multiplying it out gives you:

x^2- 5x - 20x - 100

Simplifying:

x^2 -25x - 100

There are some very important things to notice here: you get the first term of the trinomial, x ^2 (they call it a trinomial because three terms are being added) by multiplying the FIRST terms in each parenthesis (x * x).

The last number in the trinomial, the constant (meaning it has no x's in it) (100) comes from multiplying the LAST terms in each parenthesis (-20 * -5).


To get the middle term (the x term, which in this case is
-25x), first multiply the first term in the first parenthesis (x) by the last term in the second parenthesis (-5) (we can call these terms the OUTSIDE terms because they are the outside terms of the two parentheses). Then multiply the last term in the first parenthesis (-20) by the first term in the second parenthesis (x) (we can call the terms in the second parenthesis the INSIDE terms because they are the inside terms of the two parentheses). Then add the "inside" and the "outside" together. Thus the middle term
becomes -20x - 5x = -25x.

You can also remember this by remembering the word
F O I L. Foil stands for First, Outside, Inside, Last.

To review: in (x - 20)(x - 5) the First is x* x or x^2, the Outside is x * (-5) (or -5x) the Inside is -20 * x (or -20x) and the Last is (-20) * (-5) (or 100)

(x - 20) and (x - 5) are factors of x^2-25x+100.

Factoring a trinomial is just doing the reverse of what I just did in the problem above. You want to get from the trinomial back to the factors. First make sure your problem is written in the right way, that is, the term with x^2 comes first (the first term), the term with x (also called the middle term) comes next, and the constant - the term with no x in it - comes last. In your problem x^2 is the first term, -25x is the middle term, and 100 is the last term.

Step 1 is to write the problem on your paper:

x^2-25x+100

Step 2 is to write two empty parentheses like this:

( )( )

We do this because we know the factors of a trinomial look this way.

Step 3 is to look at the first term, which is x^2. We know from the problem I did above that the x^2 term comes from multiplying the first term in the first parenthesis by the first term in the second parenthesis. What are the only two things that can be multiplied together to give you x^2? That's right! x * x will give you x^2. You write it like this:

(x )(x )

Step 4 is to look at signs. Remember from the problem above that the constant term (the one with no x in it) is found by multiplying the last terms of each parenthesis. Because the sign of the constant term in your problem (100) is plus, the signs of both the numbers that multiply together to give you 100 have to be either positive or negative
(because a + * + or a - * - gives you a positive number, and a + * - will give you a negative number). So your parentheses could now look like this:

(x - )(x - ) or this: (x + )(x + )

Look back at my previous problem to see what gave you a middle term. Remember you are adding the "outside" to the "inside." If both signs are minus, this will give you a minus number; if both signs are plus it will give you a plus number. Because -25x is negative, you know that the signs have to be negative. Thus you get:

(x - )(x - )

Step 5 can be the tricky part. You know that the last term is found by multiplying the last two numbers in each parenthesis together. You also know that the inside terms
multiplied together plus the outside terms multiplied together will give you the middle term

(x-20)(x-5) FOIL

x^2-25x+100

That was a long explanation. The main things to remember are FOIL (first, outside, inside, last) and that sometimes it may take a little trial and error to get the right facts. Always multiply the factors back out to make sure you end up with what you started with.

OR

When we have a quadratic equation such as ax2+bx+c = 0, we can find out the value of x by plugging the coefficients into a formula to solve the equation. This formula is called the quadratic formula:
http://www.mste.uiuc.edu/exner/ncsa/quad/

Read this for more problems on factoring
http://mathforum.org/dr.math/faq/faq.learn.factor.html

Answer 2

you can use the formula way

you make it equal to 0, so:
x^2-25x+100=0

then use the formula (+or- b+ (root(b^2-4ac))) divided by 2a
u should find x=20
x=5
so for the equation to be equal to 0, the only factors u can have are (x-20) and (x-5)
if x=20 the equation gives 0. If x=5 its 0 too
so x^2-25x+100= (x-20)(x-5)

Answer 3

(x-20) (x-5) - you do this by inspection. Figure out what numbers that are multiplied to produce 100 will provide a sum of -25 for the x coefficient.

The answer is -5 and -20

the general form of the equation is

(x + a) (x + b) = x^2 + (a+b)x + ab

Answer 4

Use the quadratic formula.
ax^2 + bx + c = 0 is solved by

(-b +/- sqrt(b^2 - 4 * a * c))/2

a = 1
b = -25
c = 100

(25 +/- sqrt(625 - 400))/2
(25 +/- sqrt(225))/2
(25 +/- 15)/2
40/2 or 10/2
x = 20 or 5

Also expressed as

(x - 20)(x - 5) = 0

Answer 5

x^2-25x+100
(x-5)(x-20)

Answer 6

you know you need factors of 100 that add to -25.
20*5 = 100
since the middle term is negative, both factors must be negative (-20, -5)

which means:
(x-20)(x-5)

Answer 7

(x-20)(x-5) is how you factor it

Going the other way -

(x-20)(x-5)
x^2-5x-20x+100
x^2-25x+100

booya there you have it.

Answer 8

(x-20)(x-5)

That is what it is factored to.
20 x 5 = 100
you can foil it out, and you will find that my answer is correct

Answer 9

(x-20)(x-5)

Answer 10

(x-20)(x-5)=0
x-20=0
x-5=0
x=20 or x=5