How to factor 6w^2 - 5wt - 4t^2

  • Thread starter zeion
  • Start date
In summary, to factor the expression 6w^2 - 5wt - 4t^2, you can use the ac method where you find the factors of the product of the coefficients of the first and last terms that add up to the middle coefficient. In this case, the factors of 24 (the product of 6 and -4) that add up to -5 are 3 and -8, which allows you to factor the expression into (3w - 4t)(2w + t). This method can also be used for quadratics with a leading coefficient other than 1, but it may not always work if the factors of the product are irrational or complex numbers.
  • #1
zeion
466
1

Homework Statement



Factor [tex]6w^2 - 5wt - 4t^2[/tex]

Homework Equations





The Attempt at a Solution



[tex]= 6w^2 + 3wt - 8wt - 4t^2[/tex]
[tex]= 3w(2w + t) - 4t(2w + t) [/tex]
[tex]= (3w - 4t)(2w + t) [/tex]

I was able to figure out what to use for splitting up the -5wt after a lot of trial and errors, but I was just wondering if there was a general rule for finding out what to use to factor things in this form?

I noticed that I needed them to be opposite signs and is half of the coefficient of the first term and twice of the last.
 
Physics news on Phys.org
  • #2
For quadratics where the leading coefficient isn't 1, you'll often use the ac method where you multiply a and c, then find the factors of the result that add up to b. Notice that you split up the middle coefficient -5 into 3 and -8, which are two factors of 24, or the product of 6 and -4.

If that didn't make sense, just take a look on Google or read up on it http://people.richland.edu/james/misc/acmeth.html" .
 
Last edited by a moderator:
  • #3
Is there a proof for this method?
 
  • #4
If you're given some quadratic ax2+bx+c it can be proven that it can't always be factorized into the form [itex]a(x-\alpha)(x-\beta)[/itex] where [itex]\alpha[/itex] and [itex]\beta[/itex] are rational numbers, which is why factorizing can only be done through trial and error, because if they are either irrational or complex then the usual factorizing process won't work and you need to use the quadratic formula.
 

Related to How to factor 6w^2 - 5wt - 4t^2

What is factoring?

Factoring is the process of breaking down an expression into its individual factors. This can help simplify the expression and make it easier to work with.

How do I factor an expression?

To factor an expression, you need to look for common factors, such as numbers, variables, or terms. Then, use techniques like the distributive property or grouping to break down the expression into its factors.

What is the process for factoring 6w^2 - 5wt - 4t^2?

The first step is to look for common factors. In this expression, there are no common numbers, but there is a common variable, w. This can be factored out, leaving us with (6w - 5t)w - 4t^2. Then, we can use the distributive property to break down the first term further, giving us 6w^2 - 5wt - 4t^2, which is the factored form of the expression.

What are the factors of 6w^2 - 5wt - 4t^2?

The factors of 6w^2 - 5wt - 4t^2 are (6w - 5t) and (w - 4t). These are the two terms that, when multiplied together, give us the original expression.

Why is factoring important in science?

In science, factoring is important because it allows us to simplify complex expressions and equations, making them easier to understand and work with. This is especially useful in physics and chemistry, where equations can involve multiple variables and terms. Factoring also helps us identify patterns and relationships between different parts of an expression, which can provide insights and aid in problem-solving.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
4
Views
1K
  • Precalculus Mathematics Homework Help
Replies
8
Views
2K
  • Precalculus Mathematics Homework Help
Replies
15
Views
3K
  • Precalculus Mathematics Homework Help
Replies
2
Views
4K
  • Precalculus Mathematics Homework Help
Replies
8
Views
2K
  • DIY Projects
Replies
5
Views
2K
  • Cosmology
Replies
7
Views
2K
  • Precalculus Mathematics Homework Help
Replies
1
Views
917
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Precalculus Mathematics Homework Help
Replies
5
Views
2K
Back
Top