Factor $x^2-24x-17280$ Quickly & Effectively

  • MHB
  • Thread starter bergausstein
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In summary, to factor $x^2-24x-17280$ quickly and effectively, we can first look at the prime factorization of the constant term and then try to find two factors that are close in value and sum to the middle term. In this case, we can factor the expression as $(x-144)(x+120)$.
  • #1
bergausstein
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help factor this out using a faster and effective way.

$x^2-24x-17280$

I know that the factored form is $(x-144)(x+120)$

when I solved this I'm having a hard time finding a product of two numbers that will give me the middle term. can you give some fast way to determine that? with much less use of repetitious trial and error. thanks!
 
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  • #2
I would first look at the prime factorization:

\(\displaystyle 17280=2^7\cdot3^3\cdot5\)

Now, we want to find two factors whose sum is $-24$, so we know the two factors will need to be close in value. So, we could try:

\(\displaystyle 17280=\left(2^3\cdot3\cdot5\right)\left(2^3\cdot3\cdot6\right)=120\cdot144\)

And this turns out to be the factors we need. :D
 

Related to Factor $x^2-24x-17280$ Quickly & Effectively

1. What is the fastest way to factor $x^2-24x-17280$?

The fastest way to factor this quadratic equation is to use the quadratic formula, which is
x = (-b ± √(b^2 - 4ac)) / 2a.
In this equation, a = 1, b = -24, and c = -17280. Simply plug these values into the formula to find the two roots of the equation.

2. Can this equation be factored by grouping?

No, this equation cannot be factored by grouping since it does not have a common factor in all terms.

3. Is there a pattern or shortcut to factor this equation?

Yes, this equation can be factored using the "ac" method. We first need to find two numbers that multiply to get the constant term (in this case, -17280) and add up to get the coefficient of the middle term (in this case, -24). In this case, the numbers are -180 and 96. We can then rewrite the original equation as x^2-180x+96x-17280. Next, we can factor out x from the first two terms and 96 from the last two terms to get x(x-180)+96(x-180). This can be further simplified to (x+96)(x-180).

4. Are there any other methods to factor this equation?

Yes, this equation can also be factored using the "difference of squares" method. This method is useful when the equation has a squared term and a constant term, with no middle term. In this case, we can rewrite the equation as (x^2-17280) - 24x. We can then factor out a common factor of x to get x(x-24) and use the difference of squares formula to get (x+132)(x-156).

5. How can I check if my factoring is correct?

You can check if your factoring is correct by multiplying the factors back together to see if they result in the original equation. For example, if you factor x^2-24x-17280 as (x+96)(x-180), you can multiply (x+96)(x-180) using the FOIL method to get x^2-84x-17280, which is not the original equation. This means the factoring is incorrect. However, if you factor x^2-24x-17280 as (x+132)(x-156), you can multiply (x+132)(x-156) to get x^2-24x-17280, which means the factoring is correct.

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