Factor The Following Expression Completely:$\[ 5y^2 - 11y - 12 = \\]

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of simpler expressions. In this article, we will focus on factoring the given expression $5y^2 - 11y - 12$ completely. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The given expression $5y^2 - 11y - 12$ is a quadratic expression in the variable $y$.

Factoring Quadratic Expressions

There are several methods to factor quadratic expressions, including:

• Factoring by Grouping: This method involves factoring the quadratic expression by grouping the terms in pairs.
• Factoring by Difference of Squares: This method involves factoring the quadratic expression as a difference of squares.
• Factoring by Perfect Square Trinomials: This method involves factoring the quadratic expression as a perfect square trinomial.

Factoring the Given Expression

To factor the given expression $5y^2 - 11y - 12$, we will use the method of factoring by grouping.

Step 1: Factor out the Greatest Common Factor (GCF)

The first step in factoring the given expression is to factor out the greatest common factor (GCF) of the terms. In this case, the GCF is 1, so we cannot factor out any common factor.

Step 2: Group the Terms

The next step is to group the terms in pairs. We can group the terms as follows:

$5y^2 - 11y - 12 = (5y^2 - 12) - 11y$

Step 3: Factor the First Group

Now, we can factor the first group, $5y^2 - 12$, as a difference of squares:

$5y^2 - 12 = (5y - \sqrt{12})(5y + \sqrt{12})$

However, we can simplify the expression further by factoring out the greatest common factor of the terms in the first group:

$5y^2 - 12 = 5y^2 - 4 \cdot 3 = (5y - 2\sqrt{3})(5y + 2\sqrt{3})$

Step 4: Factor the Second Group

Now, we can factor the second group, $-11y$, as a single term:

$-11y = -11y$

Step 5: Combine the Factors

Finally, we can combine the factors from the first and second groups:

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y + 2\sqrt{3}) - 11y$

However, we can simplify the expression further by combining the factors:

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y + 2\sqrt{3} - \frac{11}{5})$

Step 6: Simplify the Expression

Now, we can simplify the expression further by combining the terms in the second factor:

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y + 2\sqrt{3} - \frac{11}{5})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

$5y^2 - 11y - 12 = (5y - 2\sqrt{3})(5y - \frac{11}{5} + 2\sqrt{3})$

Q&A: Factoring Quadratic Expressions

Q: What is factoring a quadratic expression?

A: Factoring a quadratic expression involves expressing it as a product of simpler expressions. In other words, it involves breaking down a quadratic expression into its prime factors.

Q: Why is factoring a quadratic expression important?

A: Factoring a quadratic expression is important because it allows us to solve quadratic equations, which are equations of the form $ax^2 + bx + c = 0$. By factoring the quadratic expression, we can set each factor equal to zero and solve for the variable.

Q: What are the different methods of factoring quadratic expressions?

A: There are several methods of factoring quadratic expressions, including:

• Factoring by Grouping: This method involves factoring the quadratic expression by grouping the terms in pairs.
• Factoring by Difference of Squares: This method involves factoring the quadratic expression as a difference of squares.
• Factoring by Perfect Square Trinomials: This method involves factoring the quadratic expression as a perfect square trinomial.

Q: How do I factor a quadratic expression using the method of factoring by grouping?

A: To factor a quadratic expression using the method of factoring by grouping, follow these steps:

1. Group the terms: Group the terms in pairs.
2. Factor the first group: Factor the first group of terms.
3. Factor the second group: Factor the second group of terms.
4. Combine the factors: Combine the factors from the first and second groups.

Q: How do I factor a quadratic expression using the method of factoring by difference of squares?

A: To factor a quadratic expression using the method of factoring by difference of squares, follow these steps:

1. Identify the difference of squares: Identify the difference of squares in the quadratic expression.
2. Factor the difference of squares: Factor the difference of squares.
3. Simplify the expression: Simplify the expression by combining the factors.

Q: How do I factor a quadratic expression using the method of factoring by perfect square trinomials?

A: To factor a quadratic expression using the method of factoring by perfect square trinomials, follow these steps:

1. Identify the perfect square trinomial: Identify the perfect square trinomial in the quadratic expression.
2. Factor the perfect square trinomial: Factor the perfect square trinomial.
3. Simplify the expression: Simplify the expression by combining the factors.

Q: What are some common mistakes to avoid when factoring quadratic expressions?

A: Some common mistakes to avoid when factoring quadratic expressions include:

• Not grouping the terms correctly: Not grouping the terms correctly can lead to incorrect factors.
• Not factoring the difference of squares correctly: Not factoring the difference of squares correctly can lead to incorrect factors.
• Not factoring the perfect square trinomial correctly: Not factoring the perfect square trinomial correctly can lead to incorrect factors.

Q: How can I practice factoring quadratic expressions?

A: You can practice factoring quadratic expressions by:

• Solving quadratic equations: Solving quadratic equations can help you practice factoring quadratic expressions.
• Factoring quadratic expressions: Factoring quadratic expressions can help you practice factoring quadratic expressions.
• Using online resources: Using online resources, such as factoring calculators and worksheets, can help you practice factoring quadratic expressions.

Conclusion

Factoring quadratic expressions is an important concept in algebra that involves expressing a quadratic expression as a product of simpler expressions. By understanding the different methods of factoring quadratic expressions, you can solve quadratic equations and simplify complex expressions. Remember to practice factoring quadratic expressions regularly to become proficient in this skill.