Transform the Fraction Equation: Simplifying 1 + y/x + x/4y = 0

Q: Why do we multiply by 4xy instead of just 4?

We need 4xy because it's the LCD that clears both fractions at once. The term $ \frac{y}{x} $ needs the x, and $ \frac{x}{4y} $ needs the 4y.

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

Look for the pattern $ a^2 + 2ab + b^2 $! Here we have $ x^2 + 4xy + 4y^2 $, which matches with a = x and b = 2y, giving us $ (x+2y)^2 $.

Q: Can I solve this problem differently?

Yes! You could substitute specific values and check each answer choice, but recognizing the algebraic pattern is faster and shows deeper understanding of perfect square trinomials.

Q: Why is the final form better than the original?

The form $ (x+2y)^2 = 0 $ is reduced because it clearly shows the relationship between x and y in its simplest factored form, making it easier to find solutions.

Perfect Square Trinomials with Fractional Equations

Consider the following relationship between x and y:

$1+\frac{y}{x}+\frac{x}{4y}=0$

Express the equation in the form of a reduced multiplication formula.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Display the equation as a shortened multiplication formula

00:04 Multiply by the common denominator to eliminate fractions

00:31 Use the commutative law and arrange the equation

00:39 Factor the expression

00:49 Use the shortened multiplication formulas to find the parentheses

00:57 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

Consider the following relationship between x and y:

$1+\frac{y}{x}+\frac{x}{4y}=0$

Express the equation in the form of a reduced multiplication formula.

Step-by-step solution

To solve this problem, let's work through these steps:

Step 1: Start with the given equation: $1+\frac{y}{x}+\frac{x}{4y}=0$ .
Step 2: Clear the fractions by multiplying all terms by $4xy$ , the common denominator.
Step 3: This gives us: $4xy + 4y^2 + x^2 = 0$ .
Step 4: Rearrange the terms for clarity: $x^2 + 4xy + 4y^2 = 0$ .
Step 5: Recognize this as the perfect square: $(x + 2y)^2 = 0$ .

This simplification results in the equation: $(x+2y)^2 = 0$ .

Therefore, the solution to the problem is $(x+2y)^2 = 0$ .

Final Answer

$(x+2y)^2=0$

Key Points to Remember

Essential concepts to master this topic

Clear Fractions: Multiply all terms by common denominator 4xy
Recognize Pattern: $x^2 + 4xy + 4y^2$ becomes $(x+2y)^2$
Verify: Expand $(x+2y)^2 = x^2 + 4xy + 4y^2 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Only multiplying some terms by the common denominator
Don't multiply just $\frac{y}{x}$ by 4xy while leaving other terms unchanged = incorrect equation! This creates an unbalanced equation with mixed fractions and whole numbers. Always multiply every single term by the same common denominator 4xy.

Practice Quiz

Test your knowledge with interactive questions

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

$$ (x+y)^2 $$

FAQ

Everything you need to know about this question

Why do we multiply by 4xy instead of just 4?

We need 4xy because it's the LCD that clears both fractions at once. The term $\frac{y}{x}$ needs the x, and $\frac{x}{4y}$ needs the 4y.

How do I recognize this as a perfect square?

Look for the pattern $a^2 + 2ab + b^2$ ! Here we have $x^2 + 4xy + 4y^2$ , which matches with a = x and b = 2y, giving us $(x+2y)^2$ .

What does it mean when we get (x+2y)² = 0?

Since a square can only equal zero when the expression inside equals zero, this means x + 2y = 0, or equivalently x = -2y.

Can I solve this problem differently?

Yes! You could substitute specific values and check each answer choice, but recognizing the algebraic pattern is faster and shows deeper understanding of perfect square trinomials.

Why is the final form better than the original?

The form $(x+2y)^2 = 0$ is reduced because it clearly shows the relationship between x and y in its simplest factored form, making it easier to find solutions.

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