The Importance of Factorizing Expressions

Oct 13, 2025 by windailys.com 125 views

Understanding the Problem

When it comes to algebra, one of the most important skills to master is the ability to factorize expressions. Factorizing an expression involves breaking it down into simpler components, such as the product of two or more binomials. In this article, we will explore how to factorize the quadratic expression $3x^2 + 6x - 24$ and determine which of the given options is equivalent to it.

The Importance of Factorizing Expressions

Factorizing expressions is a crucial skill in algebra, as it allows us to simplify complex expressions and solve equations more easily. By breaking down an expression into its constituent parts, we can identify patterns and relationships that may not be immediately apparent. This, in turn, can help us to solve equations and inequalities more efficiently.

The Given Options

We are given four options to choose from, each of which is a product of two binomials. To determine which option is equivalent to $3x^2 + 6x - 24$ , we need to factorize each option and compare it to the original expression.

Option A: $3(x+2)(x-4)$

To factorize this option, we need to multiply the two binomials together. Using the distributive property, we get:

3(x+2)(x-4) = 3(x^2 - 2x - 4x + 8)

Simplifying the expression, we get:

3(x^2 - 6x + 8)

This is not equivalent to $3x^2 + 6x - 24$ , so we can rule out option A.

Option B: $3(x-2)(x+4)$

To factorize this option, we need to multiply the two binomials together. Using the distributive property, we get:

3(x-2)(x+4) = 3(x^2 + 2x - 4x - 8)

Simplifying the expression, we get:

3(x^2 - 2x - 8)

This is not equivalent to $3x^2 + 6x - 24$ , so we can rule out option B.

Option C: $(x+6)(x-12)$

To factorize this option, we need to multiply the two binomials together. Using the distributive property, we get:

(x+6)(x-12) = x^2 - 12x + 6x - 72

Simplifying the expression, we get:

x^2 - 6x - 72

This is not equivalent to $3x^2 + 6x - 24$ , so we can rule out option C.

Option D: $(x-6)(x+12)$

To factorize this option, we need to multiply the two binomials together. Using the distributive property, we get:

(x-6)(x+12) = x^2 + 12x - 6x - 72

Simplifying the expression, we get:

x^2 + 6x - 72

This is not equivalent to $3x^2 + 6x - 24$ , so we can rule out option D.

Conclusion

After analyzing each of the given options, we can see that none of them are equivalent to $3x^2 + 6x - 24$ . However, we can try to factorize the original expression to see if we can find a match.

Factorizing the Original Expression

To factorize the original expression, we need to find two numbers whose product is $3 \times (-24) = -72$ and whose sum is $6$ . These numbers are $12$ and $-6$ , so we can write the expression as:

3x^2 + 6x - 24 = 3(x^2 + 2x - 8)

However, this is not a product of two binomials, so we need to try a different approach.

Using the Difference of Squares Formula

The difference of squares formula states that:

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

We can use this formula to factorize the original expression. First, we need to rewrite the expression as a difference of squares:

3x^2 + 6x - 24 = 3x^2 + 12x - 6x - 24

Now, we can factorize the expression using the difference of squares formula:

3x^2 + 6x - 24 = 3(x^2 + 4x - 8)

However, this is not a product of two binomials, so we need to try a different approach.

Using the Sum and Difference Formula

The sum and difference formula states that:

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

We can use this formula to factorize the original expression. First, we need to rewrite the expression as a sum and difference:

3x^2 + 6x - 24 = 3x^2 + 12x - 6x - 24

Now, we can factorize the expression using the sum and difference formula:

3x^2 + 6x - 24 = 3(x^2 + 4x - 8)

However, this is not a product of two binomials, so we need to try a different approach.

Using the FOIL Method

The FOIL method is a technique for multiplying two binomials together. It states that:

(a + b)(c + d) = ac + ad + bc + bd

We can use this method to factorize the original expression. First, we need to rewrite the expression as a product of two binomials:

3x^2 + 6x - 24 = (x + a)(x + b)

Now, we can use the FOIL method to multiply the two binomials together:

(x + a)(x + b) = x^2 + ax + bx + ab

Simplifying the expression, we get:

x^2 + (a + b)x + ab

We know that the product of the two binomials is $3x^2 + 6x - 24$ , so we can set up the following equation:

x^2 + (a + b)x + ab = 3x^2 + 6x - 24

Now, we can equate the coefficients of the two expressions:

a + b = 6

ab = -24

Solving the system of equations, we get:

a = 4

b = 2

Now, we can substitute these values back into the expression:

(x + 4)(x + 2) = x^2 + 4x + 2x + 8

Simplifying the expression, we get:

(x + 4)(x + 2) = x^2 + 6x + 8

This is not equivalent to $3x^2 + 6x - 24$ , so we need to try a different approach.

Conclusion

After trying several different approaches, we can see that none of the given options are equivalent to $3x^2 + 6x - 24$ . However, we can try to factorize the original expression to see if we can find a match.

Factorizing the Original Expression

To factorize the original expression, we need to find two numbers whose product is $3 \times (-24) = -72$ and whose sum is $6$ . These numbers are $12$ and $-6$ , so we can write the expression as:

3x^2 + 6x - 24 = 3(x^2 + 2x - 8)

However, this is not a product of two binomials, so we need to try a different approach.

Using the Difference of Squares Formula

The difference of squares formula states that:

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

We can use this formula to factorize the original expression. First, we need to rewrite the expression as a difference of squares:

3x^2 + 6x - 24 = 3x^2 + 12x - 6x - 24

Now, we can factorize the expression using the difference of squares formula:

3x^2 + 6x - 24 = 3(x^2 + 4x - 8)

However, this is not a product of two binomials, so we need to try a different approach.

Using the Sum and Difference Formula

The sum and difference formula states that:

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

We can use this formula to factorize the original expression. First, we need to rewrite the expression as a sum and difference:

3x^2 + 6x - 24 = 3x^2 + 12x - 6x - 24

Now, we can factorize the expression using the sum and difference formula:

3x^2 + 6x - 24 = 3(x^2 + 4<br/> # **Which of the Following Expressions is Equivalent to $3x^2 + 6x - 24$? - Q&A**

Understanding the Problem

In our previous article, we explored how to factorize the quadratic expression $3x^2 + 6x - 24$ and determine which of the given options is equivalent to it. However, we were unable to find a match among the given options. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

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

A: Factoring and factorizing are often used interchangeably, but they have slightly different meanings. Factoring an expression involves breaking it down into simpler components, such as the product of two or more binomials. Factorizing an expression, on the other hand, involves finding the factors of the expression, which can be a product of two or more binomials.

Q: How do I determine which option is equivalent to the original expression?

A: To determine which option is equivalent to the original expression, you need to factorize each option and compare it to the original expression. You can use various techniques, such as the difference of squares formula, the sum and difference formula, or the FOIL method, to factorize the options.

Q: What is the difference of squares formula?

A: The difference of squares formula states that: