Factoring Quadratics: Find Zeros Of 9x^2 - 63x - 702
Unlocking the secrets of quadratic expressions can feel like cracking a code, and one of the most fascinating parts is finding the zeros of a function. In this article, we're going to dive deep into the quadratic expression 9x^2 - 63x - 702. We'll explore how to find its zeros by factoring, breaking down each step in a way that's easy to understand. So, grab your math toolkit, and let's get started!
Understanding Quadratic Expressions and Zeros
Before we jump into factoring, let's make sure we're all on the same page about what quadratic expressions and zeros are. A quadratic expression is a polynomial expression of the form ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. Our expression, 9x^2 - 63x - 702, fits this description perfectly. The zeros of a function, also known as roots or x-intercepts, are the values of x that make the function equal to zero. In other words, they are the points where the parabola (the graph of a quadratic function) intersects the x-axis. Finding these zeros is a fundamental concept in algebra and has many practical applications in fields like physics, engineering, and economics.
Why Factoring Matters
There are several ways to find the zeros of a quadratic function, such as using the quadratic formula or completing the square. However, factoring is often the quickest and most straightforward method when it's applicable. Factoring involves breaking down the quadratic expression into a product of two binomials. Once we have the factored form, we can easily find the zeros by setting each factor equal to zero and solving for x. This method not only provides the solutions but also enhances our understanding of the structure of quadratic expressions and their relationship to their zeros. Factoring is a cornerstone skill in algebra, and mastering it opens doors to more advanced mathematical concepts.
Step-by-Step Factoring of 9x^2 - 63x - 702
Now, let's roll up our sleeves and factor the given quadratic expression, 9x^2 - 63x - 702. We'll take it one step at a time, so you can follow along easily.
Step 1: Look for a Common Factor
The first thing we should always do when factoring any expression is to look for a common factor among all the terms. In our case, we have 9x^2, -63x, and -702. Notice that all three coefficients (9, -63, and -702) are divisible by 9. This means we can factor out a 9 from the entire expression. Factoring out the common factor simplifies the quadratic expression and makes it easier to work with. It's like decluttering before you start organizing – it makes the whole process smoother.
So, let's factor out the 9:
9(x^2 - 7x - 78)
Now we have a simpler quadratic expression inside the parentheses: x^2 - 7x - 78. This is much easier to factor than the original expression.
Step 2: Factor the Simpler Quadratic
Next, we need to factor the quadratic expression inside the parentheses, which is x^2 - 7x - 78. To factor a quadratic in the form x^2 + bx + c, we need to find two numbers that multiply to c (in this case, -78) and add up to b (in this case, -7). This might sound like a puzzle, and that's because it is! We're essentially trying to find the right pieces that fit together to form the factored expression.
Let's think about the factors of -78. Since the product is negative, we know one factor must be positive, and the other must be negative. Also, since the sum is -7, we know the negative factor must have a larger absolute value. We can list the pairs of factors of 78 and consider their sums:
- 1 and 78
- 2 and 39
- 3 and 26
- 6 and 13
Among these pairs, 6 and 13 look promising. If we make 13 negative, we have -13 and 6, which multiply to -78 and add up to -7. Bingo! We've found our numbers.
Now we can rewrite the quadratic expression x^2 - 7x - 78 as a product of two binomials using these numbers:
(x - 13)(x + 6)
Step 3: Write the Complete Factored Form
Don't forget the common factor we pulled out in Step 1! We need to include that in our final factored form. So, the complete factored form of 9x^2 - 63x - 702 is:
9(x - 13)(x + 6)
We've successfully factored the quadratic expression. Give yourself a pat on the back!
Finding the Zeros
Now that we have the factored form, finding the zeros is the easy part. Remember, the zeros are the values of x that make the expression equal to zero. A product is zero if and only if one or more of its factors is zero. So, we need to set each factor in our factored expression equal to zero and solve for x.
Step 1: Set Each Factor to Zero
We have three factors in our factored expression: 9, (x - 13), and (x + 6). Setting each to zero gives us:
- 9 = 0 (This is never true, so it doesn't give us a zero.)
- x - 13 = 0
- x + 6 = 0
The factor 9 doesn't involve x, so it doesn't contribute to the zeros of the function. We only need to consider the binomial factors.
Step 2: Solve for x
Now, let's solve the two simple equations we have:
- x - 13 = 0 => x = 13
- x + 6 = 0 => x = -6
So, the zeros of the quadratic function 9x^2 - 63x - 702 are 13 and -6.
Verifying the Zeros
It's always a good idea to check our work, especially in math. We can verify that 13 and -6 are indeed zeros of the function by plugging them back into the original expression and making sure the result is zero. Let's do that:
Verify x = 13:
9(13)^2 - 63(13) - 702 = 9(169) - 819 - 702 = 1521 - 819 - 702 = 0
Verify x = -6:
9(-6)^2 - 63(-6) - 702 = 9(36) + 378 - 702 = 324 + 378 - 702 = 0
Both values make the expression equal to zero, so we've verified that our zeros are correct.
Conclusion
We've successfully navigated the process of finding the zeros of the quadratic function 9x^2 - 63x - 702 by factoring. We started by understanding the basics of quadratic expressions and zeros, then we broke down the factoring process into manageable steps: looking for a common factor, factoring the simpler quadratic, and writing the complete factored form. Finally, we used the factored form to easily find the zeros and verified our solutions. Factoring quadratics is a powerful tool in algebra, and mastering it will serve you well in your mathematical journey.
Remember, practice makes perfect. The more you factor quadratic expressions, the more comfortable and confident you'll become. So, keep practicing, keep exploring, and keep unlocking the secrets of mathematics!
For further reading and practice on factoring quadratic equations, check out resources like Khan Academy's Quadratic Equations section.
