Compare Expressions: Finding the Sign Between 3-2√3a+a² and (√3-a)²

Q: How do I recognize this is a perfect square?

Look for the pattern $ a^2 - 2ab + b^2 $! Here, $ a^2 $ is the $ a^2 $ term, $ b^2 $ is the constant 3 (so $ b = \sqrt{3} $), and the middle term $ -2\sqrt{3}a $ equals $ -2ab $.

Q: What if I expand the right side incorrectly?

Use the formula $ (x-y)^2 = x^2 - 2xy + y^2 $ carefully! For $ (\sqrt{3}-a)^2 $, you get $ (\sqrt{3})^2 - 2(\sqrt{3})(a) + a^2 = 3 - 2\sqrt{3}a + a^2 $.

Q: Why are these expressions equal instead of greater or less than?

Because they're identical! The left expression is already in expanded form, while the right is in factored form. When you expand $ (\sqrt{3}-a)^2 $, you get exactly the same expression.

Q: Does the value of 'a' matter for this comparison?

No! Since the expressions are algebraically identical, they're equal for any value of a. This is an identity, not an equation to solve.

Q: How can I double-check this is correct?

Pick any value for $ a $ (like $ a = 1 $) and calculate both sides. You should get the same number! For $ a = 1 $: both sides equal $ 3 - 2\sqrt{3} + 1 $.

Algebraic Identity with Perfect Square Recognition

Fill in the corresponding sign

$3-2\sqrt{3}a+a^2?(\sqrt{3}-a)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the appropriate sign

00:07 We will use short multiplication formulas to open the parentheses

00:25 The square root of a number squared equals the number itself

00:34 We notice that the expressions are equal

00:38 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

Fill in the corresponding sign

$3-2\sqrt{3}a+a^2?(\sqrt{3}-a)^2$

Step-by-step solution

The solution to the comparison problem is $=$ .

Final Answer

$=$

Key Points to Remember

Essential concepts to master this topic

Identity Recognition: Recognize $3-2\sqrt{3}a+a^2$ as perfect square form
Expansion: $(\sqrt{3}-a)^2 = 3 - 2\sqrt{3}a + a^2$
Verification: Both expressions equal same form, confirming equality ✓

Common Mistakes

Avoid these frequent errors

Not recognizing the perfect square pattern
Don't try to compare terms individually without expanding = missing the equality! Students often fail to see that $3-2\sqrt{3}a+a^2$ matches the pattern $b^2-2ab+a^2$ . Always expand $(\sqrt{3}-a)^2$ completely to reveal identical expressions.

Practice Quiz

Test your knowledge with interactive questions

$$ (4b-3)(4b-3) $$

Rewrite the above expression as an exponential summation expression:

FAQ

Everything you need to know about this question

How do I recognize this is a perfect square?

Look for the pattern $a^2 - 2ab + b^2$ ! Here, $a^2$ is the $a^2$ term, $b^2$ is the constant 3 (so $b = \sqrt{3}$ ), and the middle term $-2\sqrt{3}a$ equals $-2ab$ .

What if I expand the right side incorrectly?

Use the formula $(x-y)^2 = x^2 - 2xy + y^2$ carefully! For $(\sqrt{3}-a)^2$ , you get $(\sqrt{3})^2 - 2(\sqrt{3})(a) + a^2 = 3 - 2\sqrt{3}a + a^2$ .

Why are these expressions equal instead of greater or less than?

Because they're identical! The left expression is already in expanded form, while the right is in factored form. When you expand $(\sqrt{3}-a)^2$ , you get exactly the same expression.

Does the value of 'a' matter for this comparison?

No! Since the expressions are algebraically identical, they're equal for any value of a. This is an identity, not an equation to solve.

How can I double-check this is correct?

Pick any value for $a$ (like $a = 1$ ) and calculate both sides. You should get the same number! For $a = 1$ : both sides equal $3 - 2\sqrt{3} + 1$ .

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