Simplify The Expression:${ 3v^2 - 27t^2 }$

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Introduction

In this article, we will simplify the given expression $3v^2 - 27t^2$. This involves factoring out the greatest common factor (GCF) and applying the difference of squares formula to obtain a simplified expression.

Understanding the Expression

The given expression is $3v^2 - 27t^2$. To simplify this expression, we need to identify the GCF of the two terms. The GCF of $3v^2$ and $27t^2$ is $3$, as it is the largest factor that divides both terms.

Factoring Out the GCF

We can factor out the GCF of $3$ from both terms:

$$3v^2 - 27t^2 = 3(v^2 - 9t^2)$$

Applying the Difference of Squares Formula

The expression $v^2 - 9t^2$ is a difference of squares, which can be factored as:

$$(v - 3t)(v + 3t)$$

Therefore, the simplified expression is:

$$3(v^2 - 9t^2) = 3(v - 3t)(v + 3t)$$

Conclusion

In this article, we simplified the expression $3v^2 - 27t^2$ by factoring out the GCF and applying the difference of squares formula. The simplified expression is $3(v - 3t)(v + 3t)$.

Step-by-Step Solution

Here's a step-by-step solution to simplify the expression:

1. Identify the GCF of the two terms: $3v^2$ and $27t^2$. The GCF is $3$.
2. Factor out the GCF from both terms: $3v^2 - 27t^2 = 3(v^2 - 9t^2)$.
3. Identify the difference of squares: $v^2 - 9t^2$.
4. Factor the difference of squares: $(v - 3t)(v + 3t)$.
5. Substitute the factored difference of squares back into the expression: $3(v^2 - 9t^2) = 3(v - 3t)(v + 3t)$.

Example Use Case

The simplified expression $3(v - 3t)(v + 3t)$ can be used to solve problems involving quadratic equations. For example, if we have a quadratic equation in the form $ax^2 + bx + c = 0$, we can use the simplified expression to factor the quadratic expression and solve for the roots.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions like $3v^2 - 27t^2$:

• Identify the GCF of the two terms and factor it out.
• Look for differences of squares and factor them.
• Use the factored form to simplify the expression.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions like $3v^2 - 27t^2$:

• Failing to identify the GCF and factor it out.
• Not recognizing the difference of squares and factoring it.
• Not using the factored form to simplify the expression.

Conclusion

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Introduction

In our previous article, we simplified the expression $3v^2 - 27t^2$ by factoring out the GCF and applying the difference of squares formula. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions like this one.

Q&A

Q: What is the greatest common factor (GCF) of $3v^2$ and $27t^2$?

A: The GCF of $3v^2$ and $27t^2$ is $3$, as it is the largest factor that divides both terms.

Q: How do I factor out the GCF from both terms?

A: To factor out the GCF, we need to divide both terms by the GCF. In this case, we divide both terms by $3$ to get:

$$3v^2 - 27t^2 = 3(v^2 - 9t^2)$$

Q: What is the difference of squares formula?

A: The difference of squares formula is:

$$(a - b)(a + b) = a^2 - b^2$$

Q: How do I apply the difference of squares formula to simplify the expression?

A: To apply the difference of squares formula, we need to identify the difference of squares in the expression. In this case, we have:

$$v^2 - 9t^2 = (v - 3t)(v + 3t)$$

We can then substitute this back into the expression to get:

$$3(v^2 - 9t^2) = 3(v - 3t)(v + 3t)$$

Q: What are some common mistakes to avoid when simplifying expressions like $3v^2 - 27t^2$?

A: Some common mistakes to avoid when simplifying expressions like $3v^2 - 27t^2$ include:

• Failing to identify the GCF and factor it out.
• Not recognizing the difference of squares and factoring it.
• Not using the factored form to simplify the expression.

Q: How do I use the simplified expression to solve problems involving quadratic equations?

A: The simplified expression $3(v - 3t)(v + 3t)$ can be used to solve problems involving quadratic equations. For example, if we have a quadratic equation in the form $ax^2 + bx + c = 0$, we can use the simplified expression to factor the quadratic expression and solve for the roots.

Q: What are some tips and tricks to help me simplify expressions like $3v^2 - 27t^2$?

A: Some tips and tricks to help you simplify expressions like $3v^2 - 27t^2$ include:

• Identify the GCF of the two terms and factor it out.
• Look for differences of squares and factor them.
• Use the factored form to simplify the expression.

Conclusion

In conclusion, simplifying the expression $3v^2 - 27t^2$ involves factoring out the GCF and applying the difference of squares formula. By following the step-by-step solution and using the tips and tricks provided, you can simplify expressions like this one and solve problems involving quadratic equations.

Additional Resources

For more information on simplifying expressions and solving quadratic equations, check out the following resources:

• Simplifying Expressions
• Quadratic Equations
• Factoring Quadratic Expressions

Practice Problems

Try simplifying the following expressions using the techniques learned in this article:

• $2x^2 - 16y^2$
• $5z^2 - 25w^2$
• $3a^2 - 9b^2$

Answer Key

• $2(x - 4y)(x + 4y)$
• $5(z - 5w)(z + 5w)$
• $3(a - 3b)(a + 3b)$