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Factoring perfect squares: negative common factor

Sal factors -4t^2-12t-9 as -1(2t+3)^2. Created by Sal Khan and Monterey Institute for Technology and Education.

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• why can -4t^2-12t-9 only be solved by (a+b)(a+b)=a^2+2ab+b^2 and not by x^2+(a+b)x+ab?Thank you!
• because x^2+(a+b)x+ab has a constant 1 as leading coefficient.
while -4t^2-12t-9 has a -4 as leading coefficient.

we will need something like this to solve -4t^2-12t-9 :
(ax + b) (ax + b) = a^2x^2 + 2abx + b^2
• What is the exact meaning of Binomial?
• A binomial is polynomial with exactly 2 terms.
• Why is this type of a quadratic expression called a perfect square, when only the term and the last term are perfect squares?
• It's called a perfect square trinomial because it is created from squaring a binomial.
(a+b)^2 = a^2+2ab+b^2
• Why did he took the negative one out at the very beginning?
• It is much easier to factor trinomials when the leading coefficient is a positive number. It is also easier to see that the trinomial is a perfect square if the leading coefficient is a positive number. If you look at -4t^2 and -9, you aren't going to recognize them as perfect squares because a perfect square will always be positive. This is why Sal would have factored out the -1.
Hope this helps.
• The answer could also be (-2t-3)(2t+3) ? But its not a perfect square...?
• It is the same thing as (-1)(2t+3)^2 because all you did in that factorization is multiply one of the binomials by -1.
• At time in the video, he says that we know the 3 is positive because if it were negative, we would get -12t. Since the 12t is positive, we know that the 3 is positive. But since we square rooted 2t (the number the multiply 3 with to get 12t), 2t could also be positive or negative. So, wouldn't it also be true if both 3 and 2t were negative? -3 * -2t = 12t?
• Does it also work without factoring out the negative one? i got (2t-3)^2
(1 vote)
• Without removing the negative 1, the trinomial is not a perfect square. So, yes, you need to remove the negative one.

If you multiply your factors, you will find that they actually don't create: -4t^2 - 12t -9.
(2t-3)^2 = 4t^2 -6t -6t +9 = 4t^2 -12t +9
Your results have sign errors on the 1st and last term.
Hope this helps.