Simplify The Expression: 4 A 2 + 17 A B − 15 B 2 4a^2 + 17ab - 15b^2 4 A 2 + 17 Ab − 15 B 2

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us to manipulate and solve equations. One of the most common techniques used to simplify expressions is factoring. Factoring involves expressing an expression as a product of simpler expressions, called factors. In this article, we will learn how to simplify the expression $4a^2 + 17ab - 15b^2$ using factoring.

What is Factoring?

Factoring is a technique used to simplify expressions by expressing them as a product of simpler expressions. It involves finding the factors of an expression, which are the numbers or variables that, when multiplied together, give the original expression. Factoring can be used to simplify expressions with multiple terms, and it is a powerful tool for solving equations.

The Expression $4a^2 + 17ab - 15b^2$

The expression $4a^2 + 17ab - 15b^2$ is a quadratic expression, which means it is an expression that contains a squared variable. The expression has three terms: $4a^2$, $17ab$, and $-15b^2$. To simplify this expression, we need to find the factors of each term.

Factoring the Expression

To factor the expression $4a^2 + 17ab - 15b^2$, we need to find two numbers whose product is $-60$ (the product of the coefficients of the terms) and whose sum is $17$ (the coefficient of the middle term). These numbers are $20$ and $-3$, because $20 \times -3 = -60$ and $20 + (-3) = 17$.

Breaking Down the Expression

Now that we have found the numbers $20$ and $-3$, we can break down the expression into two parts: $4a^2 + 20ab$ and $-3ab - 15b^2$. We can then factor out the greatest common factor (GCF) from each part.

Factoring Out the GCF

The GCF of $4a^2 + 20ab$ is $4a$, and the GCF of $-3ab - 15b^2$ is $-3b$. We can factor out these GCFs to get:

$4a^2 + 20ab = 4a(a + 5b)$

$-3ab - 15b^2 = -3b(5a + 5b)$

Combining the Factors

Now that we have factored out the GCFs, we can combine the factors to get the final simplified expression:

$4a^2 + 17ab - 15b^2 = 4a(a + 5b) - 3b(5a + 5b)$

Simplifying the Expression

We can simplify the expression further by combining like terms:

$4a(a + 5b) - 3b(5a + 5b) = (4a - 3b)(a + 5b)$

Conclusion

In this article, we learned how to simplify the expression $4a^2 + 17ab - 15b^2$ using factoring. We broke down the expression into two parts, factored out the GCFs, and combined the factors to get the final simplified expression. Factoring is a powerful tool for simplifying expressions, and it is an essential skill for solving equations.

Frequently Asked Questions

• What is factoring? Factoring is a technique used to simplify expressions by expressing them as a product of simpler expressions.
• How do I factor an expression? To factor an expression, you need to find the factors of each term and combine them to get the final simplified expression.
• What is the greatest common factor (GCF)? The GCF is the largest number or variable that divides each term of an expression without leaving a remainder.

Final Answer

The final simplified expression is $(4a - 3b)(a + 5b)$.

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us to manipulate and solve equations. One of the most common techniques used to simplify expressions is factoring. Factoring involves expressing an expression as a product of simpler expressions, called factors. In this article, we will learn how to simplify the expression $4a^2 + 17ab - 15b^2$ using factoring.

What is Factoring?

Factoring is a technique used to simplify expressions by expressing them as a product of simpler expressions. It involves finding the factors of an expression, which are the numbers or variables that, when multiplied together, give the original expression. Factoring can be used to simplify expressions with multiple terms, and it is a powerful tool for solving equations.

The Expression $4a^2 + 17ab - 15b^2$

The expression $4a^2 + 17ab - 15b^2$ is a quadratic expression, which means it is an expression that contains a squared variable. The expression has three terms: $4a^2$, $17ab$, and $-15b^2$. To simplify this expression, we need to find the factors of each term.

Factoring the Expression

To factor the expression $4a^2 + 17ab - 15b^2$, we need to find two numbers whose product is $-60$ (the product of the coefficients of the terms) and whose sum is $17$ (the coefficient of the middle term). These numbers are $20$ and $-3$, because $20 \times -3 = -60$ and $20 + (-3) = 17$.

Breaking Down the Expression

Now that we have found the numbers $20$ and $-3$, we can break down the expression into two parts: $4a^2 + 20ab$ and $-3ab - 15b^2$. We can then factor out the greatest common factor (GCF) from each part.

Factoring Out the GCF

The GCF of $4a^2 + 20ab$ is $4a$, and the GCF of $-3ab - 15b^2$ is $-3b$. We can factor out these GCFs to get:

$4a^2 + 20ab = 4a(a + 5b)$

$-3ab - 15b^2 = -3b(5a + 5b)$

Combining the Factors

Now that we have factored out the GCFs, we can combine the factors to get the final simplified expression:

$4a^2 + 17ab - 15b^2 = 4a(a + 5b) - 3b(5a + 5b)$

Simplifying the Expression

We can simplify the expression further by combining like terms:

$4a(a + 5b) - 3b(5a + 5b) = (4a - 3b)(a + 5b)$

Conclusion

In this article, we learned how to simplify the expression $4a^2 + 17ab - 15b^2$ using factoring. We broke down the expression into two parts, factored out the GCFs, and combined the factors to get the final simplified expression. Factoring is a powerful tool for simplifying expressions, and it is an essential skill for solving equations.

Frequently Asked Questions

Q: What is factoring?

A: Factoring is a technique used to simplify expressions by expressing them as a product of simpler expressions.

Q: How do I factor an expression?

A: To factor an expression, you need to find the factors of each term and combine them to get the final simplified expression.

Q: What is the greatest common factor (GCF)?

A: The GCF is the largest number or variable that divides each term of an expression without leaving a remainder.

Q: How do I find the GCF of an expression?

A: To find the GCF of an expression, you need to list the factors of each term and find the largest factor that they have in common.

Q: What is the difference between factoring and simplifying an expression?

A: Factoring involves expressing an expression as a product of simpler expressions, while simplifying an expression involves combining like terms to get a simpler expression.

Q: Can I factor an expression with a negative sign?

A: Yes, you can factor an expression with a negative sign. The negative sign will be included in the factored expression.

Q: How do I know if an expression can be factored?

A: You can check if an expression can be factored by looking for common factors among the terms.

Q: What is the final simplified expression for $4a^2 + 17ab - 15b^2$?

A: The final simplified expression for $4a^2 + 17ab - 15b^2$ is $(4a - 3b)(a + 5b)$.

Final Answer

The final simplified expression is $(4a - 3b)(a + 5b)$.