Simplify the Radical Fraction: √(2x²+4xy+2y²+(x+y)²)/(x+y)

Q: Why do I need to expand (x+y)² if it's already factored?

Expanding $ (x+y)^2 = x^2 + 2xy + y^2 $ lets you combine like terms with the other parts of the expression. Without expanding, you can't see the pattern needed for factoring!

Q: How do I know when I have a perfect square inside the radical?

Look for patterns like $ a^2 + 2ab + b^2 $ or expressions that factor as $ n(x+y)^2 $. In this problem, $ 3x^2 + 6xy + 3y^2 = 3(x+y)^2 $ is the key pattern!

Q: What if (x+y) equals zero in the denominator?

Great question! When $ x+y = 0 $, the original expression is undefined because we can't divide by zero. Always note this restriction: $ x+y ≠ 0 $.

Q: Can I cancel (x+y) terms before taking the square root?

No! You must simplify inside the radical first. Only after taking the square root and getting $ \sqrt{3} \cdot (x+y) $ can you cancel with the denominator.

Q: How do I combine 4xy and 2xy correctly?

Just add the coefficients: $ 4xy + 2xy = 6xy $. Think of it like $ 4 \text{ apples} + 2 \text{ apples} = 6 \text{ apples} $ - the variable part stays the same!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 We'll factor 4 into factors 2 and 2

00:12 We'll take out common factor 2 from the parentheses

00:26 We'll use shortened multiplication formulas to find the parentheses

00:40 We'll group factors

00:49 We'll separate each factor in the root by itself

00:57 The square root of a squared number equals the number itself

01:06 We'll reduce what we can

01:11 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

$\frac{\sqrt{2x^2+4xy+2y^2+(x+y)^2}}{(x+y)}=$

2

Step-by-step solution

To solve this problem, let's go through each step in detail.

Firstly, consider the expression inside the square root. We need to work with:

2x^2 + 4xy + 2y^2 + (x+y)^2

Start by expanding $(x+y)^2$ , which is:

(x+y)^2 = x^2 + 2xy + y^2

Insert this back into the expression:

2x^2 + 4xy + 2y^2 + x^2 + 2xy + y^2

Now combine like terms:

The $x^2$ terms add up to $3x^2$ .
The $y^2$ terms add up to $3y^2$ .
The $xy$ terms add up to $6xy$ .

The expression becomes:

3x^2 + 6xy + 3y^2

Notice that this can be factored as a perfect square:

3(x^2 + 2xy + y^2)

Recognize that $x^2 + 2xy + y^2$ is $(x+y)^2$ , so:

3(x+y)^2

Take the square root of the expression:

\sqrt{3(x+y)^2} = \sqrt{3} \cdot (x+y)

The original expression under the square root now simplifies, and dividing by $(x+y)$ :

\frac{\sqrt{3} \cdot (x+y)}{(x+y)}

Cancel the common factor $(x+y)$ from numerator and denominator, leaving:

\sqrt{3}

Provided $x+y \neq 0$ , the simplified value of the original expression is:

Therefore, the solution to the problem is $\sqrt{3}$ .

3

Final Answer

$\sqrt{3}$

Key Points to Remember

Essential concepts to master this topic

Factoring: Look for perfect square patterns inside the radical expression
Technique: Combine like terms first: $2x^2 + x^2 = 3x^2$
Check: Verify by expanding $\sqrt{3}(x+y)$ back to original form ✓

Common Mistakes

Avoid these frequent errors

Forgetting to expand (x+y)² before combining terms
Don't leave (x+y)² unexpanded and try to combine it with other terms = you'll miss the perfect square pattern! This prevents proper factoring and leads to incorrect simplification. Always expand all squared terms first, then combine like terms.

Practice Quiz

Test your knowledge with interactive questions

Determine if the simplification shown below is correct:

$\frac{7}{7\cdot8}=8$

FAQ

Everything you need to know about this question

Why do I need to expand (x+y)² if it's already factored?

+

Expanding $(x+y)^2 = x^2 + 2xy + y^2$ lets you combine like terms with the other parts of the expression. Without expanding, you can't see the pattern needed for factoring!

How do I know when I have a perfect square inside the radical?

+

Look for patterns like $a^2 + 2ab + b^2$ or expressions that factor as $n(x+y)^2$ . In this problem, $3x^2 + 6xy + 3y^2 = 3(x+y)^2$ is the key pattern!

What if (x+y) equals zero in the denominator?

+

Great question! When $x+y = 0$ , the original expression is undefined because we can't divide by zero. Always note this restriction: $x+y ≠ 0$ .

Can I cancel (x+y) terms before taking the square root?

+

No! You must simplify inside the radical first. Only after taking the square root and getting $\sqrt{3} \cdot (x+y)$ can you cancel with the denominator.

How do I combine 4xy and 2xy correctly?

+

Just add the coefficients: $4xy + 2xy = 6xy$ . Think of it like $4 \text{ apples} + 2 \text{ apples} = 6 \text{ apples}$ - the variable part stays the same!

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Simplify the Radical Fraction: √(2x²+4xy+2y²+(x+y)²)/(x+y)

Radical Simplification with Perfect Square Factoring

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Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to expand (x+y)² if it's already factored?

How do I know when I have a perfect square inside the radical?

What if (x+y) equals zero in the denominator?

Can I cancel (x+y) terms before taking the square root?

How do I combine 4xy and 2xy correctly?

🌟 Unlock Your Math Potential

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