How do you factor x^2 - 22x - 48

Question

How do you factor x^2 - 22x - 48

Answer 1

x²-22x-48=0

a = 1

b = -22

c= -48

If the problem is x²-22x -48 = 0. You know you're going to be dealing with something in the format

(x -  ) (x + ).

That is because two numbers have to multiply to equal a negative integer. Only way that occurs is if one number is positive and one number is negative.

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You notice that b is a negative number (-22) so that means the absolute value of the negative number is greater than the absolute value of the positive number. If b was positive it means the absolute value of the positive number is greater than the absolute value of the negative number.

So now we know that there's going to be a positive and a negative number as well as the absolute value of the negative number is greater than the positive number.

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Now we list the factors of -48.

-1 48

-2 24

-3 16

-4 12

-6  8

-8 6

-12 4

-24 2

-48 1

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Yet because B is negative and C is negative that means the larger number must be negative so we're left with

-8 6

-12 4

-16 3

-24 2

-48 1.

Now we find the two numbers that add up to B-> =-22

-8+6 = -2 (Not what we're looking for)

-12+4= -8 (Not what we're looking for)

-16+3= -11 (Not what we're looking for)

-24+2 = -22 (Exactly what we're looking for)

-48+1= -47 (Not what we're looking for)

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We find out its -24 and 2 are the integers we're looking for. Therefore its:

(x-24)(x+2)= x²-22x-48

Answer 2

IF the original equation is:

x^2 - 22x -48  then FOIL (First, Outside, Inside, Last) would look like

(x +/- a)(x +/- b)   

NOTE:  You can use FOIL forwards (x +/- a)(x +/- b)

or backwards

x^2 (+/- ax +/- bx) +/- (a*b)

x^2 or x squared is x*x or (x)(x) is the (F)irst term

ax plus bx is the (O)utside and (I)nside added (or subtracted) together is the second term

a*b or a times b is the (L)ast term.

What do we know?

The last term is 48

Find all numbers that multipled can equal 48...(1, 48),(2, 24), (3, 16), (4, 12), (6, 8), (8, 6), (12, 4), (16, 3), (24, 2), (48, 1).

Since 48 is negative either a or b must be negative.

Which pair of the above number pairs (added or subtracted) equal the second term of -22x?

Since 22 is negative which number is negative?

So we know:

1)  that x*x= x squared....

2)  that ax +bx = -22x

3)  that a*b=-48

Once 2 and 3 above are solved...then you have the correct answer.

Joe

Answer 3

Think of all pairs of numbers that mult = last number  but add or subtract to equal  middle number

This will be easier if you Factor tree 48 = 2*2*2*2*3 

The reason you do this is that all you factors that multiply to equal 48 must come from combinations of these no.s. 

48 = 2*24  or  3*16  or  4*12  or 8*6  or 48*1

The pair that add or subtract to give middle number is  8 and 6

Now  (x        )(x        )     x and x placed in these positions always multiply to = 1st term x^2

Place 8 and 6 don't worry about the sign yet.    (x     8) (x    6)       In this position 8*6 multiply to equal last number. 

a.) If last is negative they must have opposite signs ( or we let the larger of the two #s have the sign of middle term)

b.) If last number is positive we let both signs be same and the same as the middle sign.  Thus if last sign is positive yet middle is negative then both negative.  If last is positive and middle positive then they are all positive.

If you have difficulty with these two rules you can also just do this to check your signs

(x   8)(x  6)       Mult inside terms 8*x     and Mult outside terms  x*6 place those answer underneath

       8x              You must place signs on each term so that when we are adding subtracting sum = -2x

        6x              and at the same time   8*6 still = sign of last term = -48

___________

      -2x        

(x     8) (x    6)

        -8x

         6x

___________

       -2x                   So 8 is -8  and 6 is +6      in other words   -8+2 = -6     Review your rules for adding signed numbers.

(x - 8) (x+ 6)        This is still consistent with the other rules the larger number is 8 and it gets the middle sign from -2x   of negative.

(x- 8) (x+6)   As a last step  foil to check    it is often possible to have two choices of numbers that add or subtract to give the middle term   yet only one pair is correct in that case foiling will catch those errors

x^2  - 8x  + 6x  - 48

x^2 - 2x  - 48

 

Answer 4

Hi, Kristine.

When you are factoring polynomials of this type, think of all the possible factors of the last term (neg 48):

1 * -48
2 * -24
3 * -16
4 * -12
6 * - 8
8 * - 6
12 * -4
16 * -3
24 * -2
48 * -1

Of all these pairs, one of them fits into (x         )(x       ) in such a way that the outer's and inner's from FOIL combine to give you -22x.  For example, if I use the pair 8 * -6, I would have (x + 8)(x - 6).  
FOIL'ing that back out would give me x^2 - 6x + 8x - 48.  
The outer and inner terms combine to give me a positive 2x, so this is not the right pair.

Can you find the right pair?