Question

12 select both expressions that factor $9x^2 - 30x + 25$ fully.
a $(3x + 5)(3x + 5)$
b $(3x - 5)(3x - 5)$
c $(3x + 5)^2$
d $3(x - 5)^2$
e $(3x - 5)^2$
f $(3x + 5)(3x - 5)$

Explanation:

Step1: Identify perfect square trinomial

The given expression $9x^2 - 30x + 25$ fits the form $a^2 - 2ab + b^2$, where $a=3x$ and $b=5$, since $(3x)^2=9x^2$, $2(3x)(5)=30x$, and $5^2=25$.

Step2: Factor using perfect square rule

A trinomial of the form $a^2 - 2ab + b^2$ factors to $(a-b)^2$, so $9x^2 - 30x + 25=(3x-5)^2$.

Step3: Recognize equivalent factored form

The expression $(3x-5)^2$ is equivalent to $(3x-5)(3x-5)$ (since squaring a binomial means multiplying it by itself).

Step4: Eliminate incorrect options

• Option A: $(3x+5)(3x+5)=(3x+5)^2=9x^2+30x+25$, which does not match.
• Option C: $(3x+5)^2=9x^2+30x+25$, which does not match.
• Option D: $3(x-5)^2=3(x^2-10x+25)=3x^2-30x+75$, which does not match.
• Option F: $(3x+5)(3x-5)=9x^2-25$, which does not match.

Answer:

B. $(3x - 5)(3x - 5)$
E. $(3x - 5)^2$