How do you know if # 9a^2 − 30a + 25# is a perfect square trinomial and how do you factor it?

Question

Nathan Axel · Accepted Answer

Notice that #9a^2 = (3a)^2# and #25 = 5^2#

Need to check that the middle term is #+-2*3a*5 = +-30a# - Yes.

#9a^2-30a+25 = (3a-5)^2# All perfect square trinomials are of the form: #A^2+-2AB+B^2 = (A+-B)^2# In our example, we can recognise #A=3a# and #B=5#, so just need to check that the middle term is correct. #2AB = 2 xx 3a xx 5 = 30a# So we have #A^2-2AB+B^2 = (A-B)^2 = (3a-5)^2#

Eva Abramson · Answer

undefined To determine if $9a^2 - 30a + 25$ is a perfect square trinomial, we check if its first and last terms are perfect squares and if the middle term is twice the product of the square roots of the first and last terms. Here, the first term $9a^2$ is the square of $3a$, and the last term $25$ is the square of $5$. The middle term $30a$ is twice the product of $3a$ and $5$, which is $2 \times 3a \times 5 = 30a$. Since these conditions are met, $9a^2 - 30a + 25$ is a perfect square trinomial.

To factor a perfect square trinomial, we take the square root of the first and last terms and write them as the first and last terms of the binomial. Then, we write the middle term as the product of the square roots of the first and last terms, with the same sign as the middle term in the trinomial.

So, $9a^2 - 30a + 25$ factors as $(3a - 5)^2$.