How do I come up with a middle term for an equation like x^2+99x-3870=0 in an easy way with out trial and error?

Question

how to simplify an ax^2+bx+c 

Answer 1

You could try the quadratic formula...

 

-b +/- SQRT (b^2 - 4ac)

------------------------------

              2a

 

-99 +/- SQRT (99^2 - 4*1*(-3870))

---------------------------------------------

                         2

 

-99 +/- SQRT (9801 + 15480)

-------------------------------------

                     2

 

-99 +/- SQRT (25281)

----------------------------

               2

 

-99 +/- 159

---------------

        2

 

So the roots are 30 and -129, and the factors are (x - 30) and (x + 129)

 

Answer 2

Finding the factors to the last constant as the previous examples.

In factoring...I assume you know you are finding two binomials, that when multiplied together will equal x²  +  99x  -  3870=0

        in this case...    ( x         ) ( x         ) and the factors of 3870 go on the ends

        since there is a negative, you know one of the signs has to be a negative either:

                       ( x   -   ? ) ( x    +   ? )       or       ( x   +   ? ) ( x   -   ? )

        and also since the  middle value of    x²  +  99x  -  3870    is positive   +  99x   the larger value has to be positive, because you will be subtracting both factors

    Look at both factors and subtract the factors. the difference of one of the factors should equal 99 that will be your two factors. Now just put them into the two binomials.

 

Answer 3

Note that 5*759 is not 3870, but 3795.  5*774 is the correct pair.

Answer 4

We know the factorization will result in two binomials, so let's just write the parentheses for it:

(               )(                 ) 

Since the coefficient of the first term is 1, we know that the first two terms of the binomial will simply be

(X              )(X               )

Now, the second terms of the binomial are found from finding two factors that multiply to give the last term, but also sum to give the middle term. You can just list the factors, always looking for a combination that is suspected of adding to give the middle term given some combination of signs. Knowing divisibility rules can help make this a little faster.

x² + 99x - 3870 = 0

We're looking for factors of 3870 that add together to give the middle term:

1      3870
2      1935
3      1290
5      759
6      645
9      430
10    387
15    258
18    215
30    129

30 and 129 look good. What would their signs have to be in order to sum to 99? ( -30 and +129)

So these are the other two factors.

(X - 30)(X + 129) ==> you can always FOIL this to make sure you have the right factorization.

 

Answer 5

To answer this question, it may help to look ahead to factoring quadratic trinomials (in the form ax²+bx+c).

Steps:

1. Assume a is 1. Now the equation looks like x²+bx+c.
2. if a is not 1, see note at the bottom.
3. List all the factor pairs of c. For example, if c=24 the factor pairs are:
4. 1, 24
5. 2, 12
6. 3, 8
7. 4, 6

stop when you get to the next highest square root, in this case 5=√25 (5 is not a factor of 24, so it's not included on the list)

3. Look at the SIGN (+/-) of c. If (+), then both factors have the same sign (positive times positive is a positive, negative times a negative is a positive). If (-), then the factors have opposite signs.

⇑This is "kinda" a big deal.

4, If c is (+), add the factors in each pair and compare the sum of each pair to |b| (the absolute value of b, its distance from zero. Just ignore the sign).

If c is (-), find the difference of the factors in each pair and compare this number to |b|.

(absolute value, as above)

5, Fill in the blanks (x )(x ). Ignore the operations in the middle of the parentheses for now. Order doesn't matter. For example, x²-10x+24 will factor into (check steps 3 and 4 above) (x _ 6)(x _ 4)

6, Recall the sign of c. Will the signs of the factors be the same or opposite? In the case above, the factors will have the same sign.

7, Look at the sign of b. If the factors of c have the same sign, this is the sign that fills in BOTH blanks. (above, (x-6)(x-4))

8, If the factors of c have opposite signs, the factor further away from zero wins when you add them to get b.

For example, in x²-23x-24 the sign of c is (-), so b is a difference of factors that multiply to 24. 1 and 24 have a difference of 23. This takes us to (x _ 1)(x _ 24), where the blanks will have opposite signs. The sign of b is negative, so the negative goes to the bigger factor:

(x _ 1) (x - 24). Then put the positive in the last blank (x+1)(x-24)

Back to the question:

Make factor pair lists of c. To get b, simply add the factors if c is positive, subtract them if c is negative. You can make b either sign in either case.

Answer 6

Not sure if this counts as a simple way to find the two terms, but you could solve a system of equations, i.e. 

A+B = 99

A*B= -3870

Solve the system by substitution.  I would prefer this method over trial and error.