Factor The Following Expression Completely:$x^2 - 6x - 16 = \square$

**Factor the following expression completely: $x^2 - 6x - 16 = \square$**

Understanding the Problem

The given expression is a quadratic equation in the form of $ax^2 + bx + c$. In this case, the equation is $x^2 - 6x - 16$. Our goal is to factor this expression completely, which means expressing it as a product of two binomials.

What is Factoring?

Factoring is a process of expressing an algebraic expression as a product of simpler expressions, called factors. In the case of a quadratic equation, factoring involves expressing it as a product of two binomials.

Why is Factoring Important?

Factoring is an important concept in algebra because it allows us to simplify complex expressions and solve equations more easily. By factoring an expression, we can identify its roots, which are the values of x that make the expression equal to zero.

How to Factor a Quadratic Expression

To factor a quadratic expression, we need to find two binomials whose product is equal to the original expression. The general form of a quadratic expression is $ax^2 + bx + c$. To factor it, we need to find two numbers whose product is $ac$ and whose sum is $b$.

Step-by-Step Guide to Factoring

Here's a step-by-step guide to factoring a quadratic expression:

1. Identify the coefficients: Identify the coefficients of the quadratic expression, which are the numbers in front of the $x^2$, $x$, and constant terms.
2. Find the product of the coefficients: Find the product of the coefficients, which is $ac$.
3. Find the sum of the coefficients: Find the sum of the coefficients, which is $b$.
4. Find two numbers whose product is ac and whose sum is b: Find two numbers whose product is $ac$ and whose sum is $b$.
5. Write the factored form: Write the factored form of the expression as a product of two binomials.

Example: Factoring $x^2 - 6x - 16$

Let's apply the steps above to factor the expression $x^2 - 6x - 16$.

1. Identify the coefficients: The coefficients are $a = 1$, $b = -6$, and $c = -16$.
2. Find the product of the coefficients: The product of the coefficients is $ac = 1 \times -16 = -16$.
3. Find the sum of the coefficients: The sum of the coefficients is $b = -6$.
4. Find two numbers whose product is ac and whose sum is b: We need to find two numbers whose product is $-16$ and whose sum is $-6$. The numbers are $-8$ and $2$.
5. Write the factored form: The factored form of the expression is $(x - 8)(x + 2)$.

Conclusion

Factoring a quadratic expression involves expressing it as a product of two binomials. By following the steps above, we can factor any quadratic expression. In this article, we factored the expression $x^2 - 6x - 16$ and found its factored form to be $(x - 8)(x + 2)$.

Frequently Asked Questions

Q: What is factoring? A: Factoring is a process of expressing an algebraic expression as a product of simpler expressions, called factors.

Q: Why is factoring important? A: Factoring is an important concept in algebra because it allows us to simplify complex expressions and solve equations more easily.

Q: How to factor a quadratic expression? A: To factor a quadratic expression, we need to find two binomials whose product is equal to the original expression.

Q: What are the steps to factor a quadratic expression? A: The steps to factor a quadratic expression are:

1. Identify the coefficients
2. Find the product of the coefficients
3. Find the sum of the coefficients
4. Find two numbers whose product is ac and whose sum is b
5. Write the factored form

Q: Can you give an example of factoring a quadratic expression? A: Yes, let's factor the expression $x^2 - 6x - 16$. The factored form of the expression is $(x - 8)(x + 2)$.

Q: What are the benefits of factoring? A: The benefits of factoring include simplifying complex expressions and solving equations more easily.

Q: Can you give more examples of factoring quadratic expressions? A: Yes, here are a few more examples:

• $x^2 + 5x + 6 = (x + 3)(x + 2)$
• $x^2 - 7x - 18 = (x - 9)(x + 2)$
• $x^2 + 2x - 15 = (x + 5)(x - 3)$