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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?

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5 Answers

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)

 

Comments

Comment

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.  

Comments

Nice!  rarely taught, unfortunately...

Comment

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 x+  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    x+  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.

 

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

Comments

Indeed. Thank you for that correction.

Comment

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.