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# Factoring difference of squares: leading coefficient ≠ 1

Sal factors 45x^2-125 as 5(3x+5)(3x-5). Created by Sal Khan.

## Want to join the conversation?

• Can 25 can be squared? Yes or no?
• Brian Lee
25 can be squared, and so can any number. Squared means to multiply by itself. When you are finding a square root, not every number can have a square root. There is a helpful song that i know, ahem:
1x1=1
2x2=4
3x3=9
4x4=16
5x5=25********
6x6=36
7x7=49
8x8=64
9x9=81
10x10 is a HUNDRED! EVERY BODY DANCE!
So yes, 25 can be squared and 25 squared will be 625. as i said, any number can be squared. Not every number has a square root. some examples of numbers not have ing a square root is:2,3,5,6,7,8,10, etc. Some number have a square root that is a decimal, but not all. Most do, that arent normally square rotted. Hope this helps :)
• How would you solve a difference of squares for 100g^2 -1
• Sometic,
100g² is (10g)² and
1² is (1)² so both terms are squares. So factor using the difference of squares and you have
(10g+1)(10g-1)

I hope that helps make it click for you.
• What does ^ mean?
• And on your calculator you see ^ as well, it does the same thing.
if you have INTL keyboard you can do it the right way: 3² 7³ 5²
• How do you solve these types of problems when the constant is a non-square number?
For example (and don't just solve it!):
``96−6x^2``
• First, you must divide by a common factor until both terms are perfect squares, if possible. In this problem, 96 and -6 are both divisible by 6.
After the division, we have 16-x^2. Now you can factor it in the "difference of two squares" method. The square root of 16 is 4, and the square root of x^2 is x.
Important: The factor with the subtraction/negative sign always comes second in the factored form.
Important: Don't forget the factor you divided by! It's the leading coefficient in your final answer.
Knowing this, the factored form of 96-6x^2 is `6(4+x)(4-x).`

I hope this cleared things up.
• why dint he considered sqrt(45)x to power of 2 - sqrt(125) to power of 2 ?
• He's factoring the 5 out to get perfect squares, 45 and 125 aren't perfect squares so the square root would be irrational/long decimal. Power of 2 cancels out the square root so you would be left with 45-125 after your equation.
• what is the purpose of the 5 outside the perenthesis
• What if the questions say solve it? what are x?
• How can factor 3x^2 + 10x + 8 in the forn of (3x ) (x )
or 12x^2 - 17x + 6 = (4x ) (3x )?
Is there a rule to do this? Completing the square will not do.
Where on the Academy can I find the videos for this?
I believe that it is most trial and error, but is a way for trial and least error?
Very, very,very much thanks.
Your advice may also be sent to rainer@dds.nl
Thank you, happy study on the Khan Ac.
• You can apply the consequence of Fundamental Theorem of Algebra.

Every quadratic expression of type ax^2 + bx + c can be factored by the form

a(x - x1)(x- x2), where x1 and x2 are the roots of equation.

3x^2 + 10x + 8 = 0

x1 = -4/3
x2 = -2

Then, you get a(x-x1)(x-x2) = 3(x + 4/3)(x+2) = (3x+4)(x+2)

Sorry for my English.
• Can you show us how to completely factor and factor trinomials?
• If this were too be an actual question on a test, which answer would be accepted? Or would both be acceptable? 5(9x^2 - 25) or 5(3x + 5)(3x - 5)?