Find the Missing Sign in m³+2m²n=4m²+n²: Comparing Perfect Square Expressions

Q: Why can't I just expand and compare the expressions directly?

While you can expand both expressions, the given constraint $ m^3 + 2m^2n = 4m^2 + n^2 $ creates a special relationship between m and n that limits what values they can take. Without knowing specific values, comparison isn't possible.

Q: What does 'It is not possible to calculate' actually mean?

This means we don't have enough information to determine which sign (>,

Q: Should I try substituting random values for m and n?

Be careful! You can't use just any values - they must satisfy the constraint $ m^3 + 2m^2n = 4m^2 + n^2 $. Even then, different valid pairs might give different comparison results.

Q: How do I recognize when a problem has insufficient information?

Look for these signs: Multiple variables with only one constraintQuestions asking for definitive comparisons without specific valuesComplex relationships that don't simplify clearly When in doubt, check if you can find counterexamples!

Q: Is this type of problem common in algebra?

Yes! Real-world problems often have indeterminate situations. Learning to recognize when you don't have enough information is just as important as solving problems with clear answers.

Algebraic Constraints with Indeterminate Comparisons

Replace ? with the missing sign given that $m^3+2m^2n=4m^2+n^2$ :

$m(m+n)^2?(2m+n)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the sign

00:03 We'll use shortened multiplication formulas to open the parentheses

00:24 We'll properly open parentheses and multiply by each factor

00:44 We'll compare according to the given equation

00:51 We'll reduce what we can

00:54 We'll reduce what we can

00:58 We don't know anything about N so it's impossible to know

01:03 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

Replace ? with the missing sign given that $m^3+2m^2n=4m^2+n^2$ :

$m(m+n)^2?(2m+n)^2$

Step-by-step solution

To solve this problem, we aim to relate the expression $m(m+n)^2$ and $(2m+n)^2$ given the equation $m^3 + 2m^2n = 4m^2 + n^2$ .

The key step is to understand that due to the given constraint equation, without loss of generality or additional specific values for $m$ and $n$ , a direct comparison to determine the sign $?$ is not feasible under the constraint $m^3 + 2m^2n = 4m^2 + n^2$ . The nature of the relationship defined by the constraint tells us that the relationship between these expressions might hold under special conditions, but projecting one side as definitively greater, less, or equal is not straightforward.

Given this complexity and the lack of further simplifiable information directly provided by the equation, it is not possible to calculate or visually confirm which sign should be appropriately used without more specific values or further simplification details.

Therefore, the answer is: It is not possible to calculate.

Final Answer

It is not possible to calculate.

Key Points to Remember

Essential concepts to master this topic

Constraint Analysis: Given equation limits possible relationships between expressions
Technique: Expand both sides: $m(m+n)^2 = m(m^2+2mn+n^2)$
Check: Test if constraint allows definitive comparison: insufficient information ✓

Common Mistakes

Avoid these frequent errors

Assuming you can always compare algebraic expressions
Don't try to determine >, <, or = without sufficient information = wrong conclusion! The constraint equation doesn't provide enough data to establish a definitive relationship. Always check if you have enough information before making comparisons.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

$$ (x+3)^2 $$

FAQ

Everything you need to know about this question

Why can't I just expand and compare the expressions directly?

While you can expand both expressions, the given constraint $m^3 + 2m^2n = 4m^2 + n^2$ creates a special relationship between m and n that limits what values they can take. Without knowing specific values, comparison isn't possible.

What does 'It is not possible to calculate' actually mean?

This means we don't have enough information to determine which sign (>, <, or =) is correct. The constraint equation alone doesn't tell us whether one expression is always bigger, smaller, or equal to the other.

Should I try substituting random values for m and n?

Be careful! You can't use just any values - they must satisfy the constraint $m^3 + 2m^2n = 4m^2 + n^2$ . Even then, different valid pairs might give different comparison results.

How do I recognize when a problem has insufficient information?

Look for these signs:

Multiple variables with only one constraint
Questions asking for definitive comparisons without specific values
Complex relationships that don't simplify clearly

When in doubt, check if you can find counterexamples!

Is this type of problem common in algebra?

Yes! Real-world problems often have indeterminate situations. Learning to recognize when you don't have enough information is just as important as solving problems with clear answers.

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