Solve the Fraction Equation: Finding the Numerator in ?/(25x^4-5x^2) = 3/(5x^2)

Let's examine the following problem:

$\frac{?}{25x^4-5x^2}=\frac{3}{5x^2}$

Note that in the denominator of the fraction on the left side there is an expression that can be factored using factoring out a common factor. Hence we will factor out the largest possible common factor (meaning that the expression left in parentheses cannot be further factored by taking out a common factor):

$\frac{?}{25x^4-5x^2}=\frac{3}{5x^2} \\ \downarrow\\ \frac{?}{5x^2(5x^2-1)}=\frac{3}{5x^2} \\$In factoring, we used of course the law of exponents:

$\bm{a^{m+n}=a^m\cdot a^n}$

Let's continue solving the problem. Remember the fraction reduction operation. Note that in the fraction's denominator both on the right side and on the left side there is the expression:$5x^2$ Therefore we don't want it to be reduced from the denominator on the left side. However, the expression:

$5x^2-1$,

is not found in the denominator on the right side (which is the fraction after reduction) Therefore we can conclude that this expression needs to be reduced from the denominator on the left side,

Additionally, let's consider the number 3 which appears in the numerator on the right side (which is the fraction after reduction) but is not found in the numerator on the left side, meaning - we want it to be included in choosing the missing expression (which is the product of the desired expressions - in order to obtain the fraction on the right side after reduction)

Therefore the missing expression must be none other than:

$3(5x^2-1)$

Let's verify that from this choice we obtain the expression on the right side: (reduction sign)

$\frac{?}{5x^2(5x^2-1)}=\frac{3}{5x^2} \\ \downarrow\\ \frac{\textcolor{red}{3(5x^2-1)}}{5x^2(5x^2-1)}\stackrel{?}{= }\frac{3}{5x^2} \\ \downarrow\\ \boxed{\frac{3}{5x^2} \stackrel{!}{= }\frac{3}{5x^2} }$

Therefore choosing the expression:

$3(5x^2-1)$

is indeed correct.

From opening the parentheses (using the distributive law) we can identify that the correct answer is answer A.