Factor The Expression:$3x^2 + 3x - 6$

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Introduction

Factoring an algebraic expression is a crucial step in solving equations and inequalities. It involves expressing the given expression as a product of simpler expressions, called factors. In this article, we will focus on factoring the quadratic expression $3x^2 + 3x - 6$. We will use various factoring techniques, including the greatest common factor (GCF) method, the difference of squares method, and the grouping method.

Understanding the Expression

Before we begin factoring, let's analyze the given expression $3x^2 + 3x - 6$. This is a quadratic expression, which means it has a degree of 2. The expression consists of three terms: $3x^2$, $3x$, and $-6$. We can see that the first two terms have a common factor of $3x$, while the third term is a constant.

Factoring by Greatest Common Factor (GCF)

The GCF method is a simple and effective way to factor an expression. It involves finding the greatest common factor of all the terms in the expression. In this case, the GCF of $3x^2$, $3x$, and $-6$ is $3$. We can factor out the GCF from each term:

$$3x^2 + 3x - 6 = 3(x^2 + x - 2)$$

Now, we are left with a quadratic expression inside the parentheses: $x^2 + x - 2$. We can try to factor this expression further using other factoring techniques.

Factoring by Difference of Squares

The difference of squares method is another useful technique for factoring quadratic expressions. It involves expressing the expression as the difference of two squares. In this case, we can rewrite the expression $x^2 + x - 2$ as:

$$x^2 + x - 2 = (x + \frac{1}{2})^2 - (\frac{1}{2})^2 - 2$$

However, this does not seem to help us factor the expression further. We can try another technique.

Factoring by Grouping

The grouping method is a more advanced technique for factoring quadratic expressions. It involves grouping the terms in the expression in a way that allows us to factor them. In this case, we can group the first two terms and the last term:

$$x^2 + x - 2 = (x^2 + x) - 2$$

Now, we can factor out a common factor from each group:

$$(x^2 + x) - 2 = x(x + 1) - 2$$

However, this does not seem to help us factor the expression further. We can try another technique.

Factoring by Completing the Square

The completing the square method is a technique for factoring quadratic expressions that cannot be factored using other methods. It involves rewriting the expression in a way that allows us to factor it. In this case, we can rewrite the expression $x^2 + x - 2$ as:

$$x^2 + x - 2 = (x + \frac{1}{2})^2 - (\frac{1}{2})^2 - 2$$

However, this does not seem to help us factor the expression further. We can try another technique.

Factoring by Using the Quadratic Formula

The quadratic formula is a technique for solving quadratic equations. It involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions to the equation. In this case, we can use the quadratic formula to find the solutions to the equation $x^2 + x - 2 = 0$:

$$x = \frac{-1 \pm \sqrt{1 + 8}}{2}$$

$$x = \frac{-1 \pm \sqrt{9}}{2}$$

$$x = \frac{-1 \pm 3}{2}$$

$$x = 1, -2$$

Now, we can factor the expression $x^2 + x - 2$ as:

$$(x - 1)(x + 2)$$

Conclusion

In this article, we have factored the quadratic expression $3x^2 + 3x - 6$ using various techniques, including the GCF method, the difference of squares method, the grouping method, and the completing the square method. We have also used the quadratic formula to find the solutions to the equation $x^2 + x - 2 = 0$. The final factored form of the expression is:

$$3(x - 1)(x + 2)$$

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Frequently Asked Questions

Q: What is the greatest common factor (GCF) of the expression $3x^2 + 3x - 6$?

A: The GCF of the expression $3x^2 + 3x - 6$ is $3$.

Q: How do I factor the expression $3x^2 + 3x - 6$ using the GCF method?

A: To factor the expression $3x^2 + 3x - 6$ using the GCF method, you need to factor out the GCF from each term. In this case, the GCF is $3$, so you can factor out $3$ from each term:

$$3x^2 + 3x - 6 = 3(x^2 + x - 2)$$

Q: Can I factor the expression $x^2 + x - 2$ further using other factoring techniques?

A: Yes, you can try to factor the expression $x^2 + x - 2$ further using other factoring techniques, such as the difference of squares method, the grouping method, or the completing the square method.

Q: How do I factor the expression $x^2 + x - 2$ using the difference of squares method?

A: To factor the expression $x^2 + x - 2$ using the difference of squares method, you need to rewrite the expression as the difference of two squares. However, this does not seem to help us factor the expression further.

Q: How do I factor the expression $x^2 + x - 2$ using the grouping method?

A: To factor the expression $x^2 + x - 2$ using the grouping method, you need to group the terms in the expression in a way that allows you to factor them. In this case, you can group the first two terms and the last term:

$$x^2 + x - 2 = (x^2 + x) - 2$$

Now, you can factor out a common factor from each group:

$$(x^2 + x) - 2 = x(x + 1) - 2$$

However, this does not seem to help us factor the expression further.

Q: How do I factor the expression $x^2 + x - 2$ using the completing the square method?

A: To factor the expression $x^2 + x - 2$ using the completing the square method, you need to rewrite the expression in a way that allows you to factor it. However, this does not seem to help us factor the expression further.

Q: Can I use the quadratic formula to find the solutions to the equation $x^2 + x - 2 = 0$?

A: Yes, you can use the quadratic formula to find the solutions to the equation $x^2 + x - 2 = 0$. The quadratic formula is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = 1$, and $c = -2$. Plugging these values into the quadratic formula, you get:

$$x = \frac{-1 \pm \sqrt{1 + 8}}{2}$$

$$x = \frac{-1 \pm \sqrt{9}}{2}$$

$$x = \frac{-1 \pm 3}{2}$$

$$x = 1, -2$$

Q: How do I factor the expression $x^2 + x - 2$ using the quadratic formula?

A: To factor the expression $x^2 + x - 2$ using the quadratic formula, you need to find the solutions to the equation $x^2 + x - 2 = 0$. The solutions to this equation are $x = 1$ and $x = -2$. You can factor the expression $x^2 + x - 2$ as:

$$(x - 1)(x + 2)$$

Q: What is the final factored form of the expression $3x^2 + 3x - 6$?

A: The final factored form of the expression $3x^2 + 3x - 6$ is:

$$3(x - 1)(x + 2)$$

We hope this Q&A article has provided a clear and concise explanation of how to factor the expression $3x^2 + 3x - 6$.