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### Course: Algebra (all content)>Unit 10

Lesson 18: Factoring quadratics: Perfect squares

# 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.

## Want to join the conversation?

• 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.
• 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.
• 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.
• 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?
• Can Khan Academy teach cubics?
• Can the answer be (-2t-3)^2 ?
• No.
You can try expanding (-2t - 3)². You won't get the original equation.

This is because a negative number squared is positive.
By taking the common factor (-1) between (-2t - 3), we get
(-1) * (2t + 3)
Squaring it we get
(-1)² * (2t + 3)²
= (2t + 3)²
• what does Sal mean when he says "factor by grouping" 40 second in to the video?