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

Q: Why do I need to factor the denominator first?

Factoring $ 25x^4-5x^2 = 5x^2(5x^2-1) $ reveals the common factor $ 5x^2 $ that will cancel out! Without factoring, you can't see what needs to reduce to get the simplified fraction.

Q: How do I know what the missing numerator should be?

The missing numerator must create a fraction that reduces to $ \frac{3}{5x^2} $. Since $ 5x^2 $ cancels from top and bottom, you need $ 3(5x^2-1) $ so the remaining factor $ (5x^2-1) $ also cancels.

Q: What does 'reduction' mean in fraction equations?

Reduction means canceling identical factors from numerator and denominator. Like $ \frac{6}{8} = \frac{3}{4} $ by canceling 2, here we cancel $ 5x^2 $ and $ (5x^2-1) $.

Q: Why is the answer 15x² - 3 and not 15x² + 3?

When you expand $ 3(5x^2-1) $ using the distributive property: $ 3 \cdot 5x^2 - 3 \cdot 1 = 15x^2 - 3 $. The minus sign comes from the $ -1 $ in the factored form.

Q: How can I check if my answer is correct?

Substitute your answer back: $ \frac{15x^2-3}{25x^4-5x^2} = \frac{3(5x^2-1)}{5x^2(5x^2-1)} = \frac{3}{5x^2} $ ✓. The fractions should be identical after reduction!

Q: What if I can't factor the denominator?

Look for common factors first! In $ 25x^4-5x^2 $, both terms have $ 5x^2 $ in common. Factor it out: $ 5x^2(5x^2-1) $. Practice identifying the greatest common factor!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the appropriate numerator

00:04 Let's factor 25 into factors 5 and 3, and 4 squared twice

00:18 Let's mark the common factors

00:22 Let's take out the common factors from the parentheses

00:38 We want to isolate the numerator, so we'll multiply by the denominator

01:00 Let's properly open the parentheses, multiply by each factor

01:16 Let's calculate the products

01:22 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Complete the corresponding expression in the numerator

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

2

Step-by-step solution

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.

3

Final Answer

$15x^2-3$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Factor denominators to identify common terms for reduction
Cross-Multiplication: $\frac{?}{25x^4-5x^2} = \frac{3}{5x^2}$ becomes $? \cdot 5x^2 = 3(25x^4-5x^2)$
Verification Check: Substitute answer back to ensure reduction gives $\frac{3}{5x^2}$ ✓

Common Mistakes

Avoid these frequent errors

Not factoring the denominator before solving
Don't try to solve $\frac{?}{25x^4-5x^2} = \frac{3}{5x^2}$ without factoring = messy calculations and wrong answers! Students miss that $25x^4-5x^2 = 5x^2(5x^2-1)$ , making it impossible to see the reduction pattern. Always factor polynomial denominators first to identify common terms.

Practice Quiz

Test your knowledge with interactive questions

Identify the field of application of the following fraction:

$\frac{7}{13+x}$

FAQ

Everything you need to know about this question

Why do I need to factor the denominator first?

+

Factoring $25x^4-5x^2 = 5x^2(5x^2-1)$ reveals the common factor $5x^2$ that will cancel out! Without factoring, you can't see what needs to reduce to get the simplified fraction.

How do I know what the missing numerator should be?

+

The missing numerator must create a fraction that reduces to $\frac{3}{5x^2}$ . Since $5x^2$ cancels from top and bottom, you need $3(5x^2-1)$ so the remaining factor $(5x^2-1)$ also cancels.

What does 'reduction' mean in fraction equations?

+

Reduction means canceling identical factors from numerator and denominator. Like $\frac{6}{8} = \frac{3}{4}$ by canceling 2, here we cancel $5x^2$ and $(5x^2-1)$ .

Why is the answer 15x² - 3 and not 15x² + 3?

+

When you expand $3(5x^2-1)$ using the distributive property: $3 \cdot 5x^2 - 3 \cdot 1 = 15x^2 - 3$ . The minus sign comes from the $-1$ in the factored form.

How can I check if my answer is correct?

+

Substitute your answer back: $\frac{15x^2-3}{25x^4-5x^2} = \frac{3(5x^2-1)}{5x^2(5x^2-1)} = \frac{3}{5x^2}$ ✓. The fractions should be identical after reduction!

What if I can't factor the denominator?

+

Look for common factors first! In $25x^4-5x^2$ , both terms have $5x^2$ in common. Factor it out: $5x^2(5x^2-1)$ . Practice identifying the greatest common factor!

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

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