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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 is the purpose of the 5 outside the perenthesis
• The 5 is the Greatest Common Factor that was factored from the polynomial using the distributive property. The new binomial is a difference of 2 squares, so it can be factored to another level using that technique. Thus you create 3 factors: 5 (3x - 5) (3x + 5)
• 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.
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 ?
• What does ^ mean?
(1 vote)
• 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 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.
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.
• 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)?
• That would depend on what the question asked for. If it asks for "fully factored" then 5(3x + 5)(3x - 5) would be correct.

So, you need to know the names of the various forms for expressions so that you know which form a test question is asking for.
• Would (15x+25)(15x-25) be a correct answer due to the distributive property? Would it count as simplying and be a valid answer for a question asking for simplification or would it just be... Unsimplifying again? lol
• Actually that's wrong because you multiplied both factors in parentheses by 5. If you want to distribute the 5, you can multiply it by either of the two factors. Multiplying it by both is like multiplying by 5*5 or 25 - so you just changed the equation.
• How do you factor the difference of two squares?