Simplify the Expression: m²/9 - 4mn/3 + 4n² Step-by-Step

Q: How do I know if this is a perfect square trinomial?

Look for the pattern $ a^2 - 2ab + b^2 $. If the first and last terms are perfect squares and the middle term is exactly twice their product, then it factors as $ (a-b)^2 $.

Q: Why is a = m/3 and not something else?

Since the first term is $ \frac{m^2}{9} $, we need $ a^2 = \frac{m^2}{9} $. Taking the square root: $ a = \frac{m}{3} $. This is the only value that works!

Q: How do I find b from the middle term?

The middle term is $ -\frac{4mn}{3} $, which should equal $ -2ab $. Since $ a = \frac{m}{3} $, we get: $ -2 \cdot \frac{m}{3} \cdot b = -\frac{4mn}{3} $, so b = 2n.

Q: What if I can't see the perfect square pattern?

Try working backwards! Look at each answer choice and expand them to see which one gives you the original expression. This helps you understand the pattern better.

Q: Do I always get a perfect square with three terms?

No! Not all trinomials are perfect squares. Always check that the pattern $ a^2 ± 2ab + b^2 $ matches exactly before factoring as $ (a ± b)^2 $.

Perfect Square Trinomials with Fractional Terms

$\frac{m^2}{9}-\frac{4}{3}mn+4n^2=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express as brackets

00:03 Break down 9 into 3 squared

00:11 Break down 4 into 2 squared

00:17 Extract root and square

00:29 Factor with 2 and 2

00:32 Extract root and square

00:49 Use shortened multiplication formulas to find the brackets

01:00 These are A and B

01:06 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\frac{m^2}{9}-\frac{4}{3}mn+4n^2=\text{?}$

Step-by-step solution

To solve this problem, we'll simplify the expression $\frac{m^2}{9} - \frac{4}{3}mn + 4n^2$ to reveal a possible square of a binomial:

Step 1: Recognize the expression $\frac{m^2}{9} - \frac{4}{3}mn + 4n^2$ may have terms fitting the identity $(a - b)^2 = a^2 - 2ab + b^2$ .
Step 2: Identify parts of the expression:
- $a^2 = \frac{m^2}{9}$ , so $a = \frac{m}{3}$ .
- $-2ab = -\frac{4}{3}mn$ . Since $a = \frac{m}{3}$ , solve for $b$ :
$2 \times \frac{m}{3} \times b = \frac{4}{3}mn \Rightarrow 2b = 4n \Rightarrow b = 2n$ .
Step 3: Check $b^2$ : $b^2 = (2n)^2 = 4n^2$ , which matches the $4n^2$ term in the expression.

Therefore, the correct form of the expression is:
$(\frac{m}{3} - 2n)^2$ .

Thus, the expression simplifies to:

$(\frac{m}{3} - 2n)^2$

Final Answer

$(\frac{m}{3}-2n)^2$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Look for $a^2 - 2ab + b^2 = (a-b)^2$ structure
Technique: From $\frac{m^2}{9}$ , identify $a = \frac{m}{3}$ for the first term
Check: Expand $(\frac{m}{3} - 2n)^2$ to verify it equals original expression ✓

Common Mistakes

Avoid these frequent errors

Incorrectly identifying the perfect square pattern
Don't assume any trinomial is a perfect square without checking all three terms! Students often guess patterns and get expressions like $(\frac{m}{9} - 4n)^2$ which expand incorrectly. Always verify by identifying $a$ from the first term, then check if $-2ab$ and $b^2$ match exactly.

Practice Quiz

Test your knowledge with interactive questions

Declares the given expression as a sum

$$ (7b-3x)^2 $$

FAQ

Everything you need to know about this question

How do I know if this is a perfect square trinomial?

Look for the pattern $a^2 - 2ab + b^2$ . If the first and last terms are perfect squares and the middle term is exactly twice their product, then it factors as $(a-b)^2$ .

Why is a = m/3 and not something else?

Since the first term is $\frac{m^2}{9}$ , we need $a^2 = \frac{m^2}{9}$ . Taking the square root: $a = \frac{m}{3}$ . This is the only value that works!

How do I find b from the middle term?

The middle term is $-\frac{4mn}{3}$ , which should equal $-2ab$ . Since $a = \frac{m}{3}$ , we get: $-2 \cdot \frac{m}{3} \cdot b = -\frac{4mn}{3}$ , so b = 2n.

What if I can't see the perfect square pattern?

Try working backwards! Look at each answer choice and expand them to see which one gives you the original expression. This helps you understand the pattern better.

Do I always get a perfect square with three terms?

No! Not all trinomials are perfect squares. Always check that the pattern $a^2 ± 2ab + b^2$ matches exactly before factoring as $(a ± b)^2$ .

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