Finding Zeros: Y = X^2 - 12x + 35 Explained

Hey guys! Let's dive into solving a common problem in algebra: finding the zeros of a quadratic function. Today, we're going to tackle the function y = x^2 - 12x + 35. Finding the zeros, also known as the roots or x-intercepts, means we need to figure out the values of x that make y equal to zero. In simpler terms, where does the parabola cross the x-axis? Let's get started and break this down step by step. This explanation will not only help you solve this particular problem but also equip you with the skills to handle similar quadratic equations. Understanding the underlying concepts is key, so we'll explore different methods and why they work. Whether you're a student brushing up on your algebra or just curious about math, you've come to the right place!

Understanding Quadratic Functions and Zeros

Before we jump into solving, let's make sure we're all on the same page about what a quadratic function is and what we mean by "zeros." A quadratic function is a polynomial function of the second degree, which means the highest power of x is 2. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants and a is not equal to zero. Our function, y = x^2 - 12x + 35, fits this form perfectly, with a = 1, b = -12, and c = 35.

The graph of a quadratic function is a parabola, a U-shaped curve. The zeros of the function are the points where the parabola intersects the x-axis. These are the x-values for which y (or f(x)) equals zero. A quadratic function can have two real zeros, one real zero (a repeated root), or no real zeros (meaning the parabola doesn't intersect the x-axis). Finding these zeros is crucial in many mathematical applications and real-world problems, from physics to engineering.

There are several methods we can use to find the zeros of a quadratic function, including factoring, using the quadratic formula, and completing the square. Each method has its strengths and is suitable for different types of quadratic equations. For the function y = x^2 - 12x + 35, factoring is often the quickest and easiest approach, but we'll also briefly touch on the quadratic formula to illustrate another option. By understanding these different methods, you'll be well-equipped to tackle a variety of quadratic equations.

Method 1: Factoring the Quadratic Function

Okay, let's get our hands dirty and find the zeros by factoring. Factoring is a technique where we rewrite the quadratic expression as a product of two binomials. This method works best when the quadratic expression can be easily factored, which is the case for our function, y = x^2 - 12x + 35. The key idea behind factoring is to reverse the process of expanding two binomials using the FOIL (First, Outer, Inner, Last) method.

Here's how we can factor x^2 - 12x + 35: We need to find two numbers that multiply to 35 (the constant term) and add up to -12 (the coefficient of the x term). Think about the factors of 35: 1 and 35, and 5 and 7. Since we need a negative sum, we'll consider negative factors. The pair -5 and -7 works perfectly because (-5) * (-7) = 35 and (-5) + (-7) = -12. So, we can rewrite the quadratic expression as:

x^2 - 12x + 35 = (x - 5)(x - 7)

Now that we've factored the quadratic, we can find the zeros by setting each factor equal to zero and solving for x. This is based on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, we have:

x - 5 = 0 or x - 7 = 0

Solving these equations gives us:

x = 5 or x = 7

Therefore, the zeros of the quadratic function y = x^2 - 12x + 35 are 5 and 7. This means the parabola intersects the x-axis at the points (5, 0) and (7, 0). Factoring is a powerful tool, and with practice, you'll be able to quickly factor many quadratic expressions. It's like having a secret code to unlock the solutions!

Method 2: Using the Quadratic Formula

Now, let's explore another method for finding the zeros of a quadratic function: the quadratic formula. This formula is a universal tool that works for any quadratic equation, regardless of whether it's easily factorable or not. It's a bit like having a Swiss Army knife for quadratic equations – it can handle anything! The quadratic formula is derived from the process of completing the square and is a fundamental concept in algebra.

The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0. In our case, for the function y = x^2 - 12x + 35, we have a = 1, b = -12, and c = 35. Let's plug these values into the formula and see what we get:

x = (-(-12) ± √((-12)^2 - 4 * 1 * 35)) / (2 * 1)

Simplifying this, we have:

x = (12 ± √(144 - 140)) / 2 x = (12 ± √4) / 2 x = (12 ± 2) / 2

This gives us two possible solutions:

x = (12 + 2) / 2 = 14 / 2 = 7 x = (12 - 2) / 2 = 10 / 2 = 5

As you can see, we arrived at the same zeros as before: 5 and 7. The quadratic formula is a reliable method, especially when factoring is difficult or impossible. It's essential to memorize this formula and understand how to apply it correctly. While it might seem a bit intimidating at first, with practice, it becomes a straightforward process. Plus, knowing the quadratic formula gives you a powerful tool in your mathematical arsenal!

Verifying the Zeros and Graphing

Great! We've found the zeros of the quadratic function y = x^2 - 12x + 35 using both factoring and the quadratic formula. But let's take a moment to verify our results and visualize what's happening. This step is crucial to ensure accuracy and deepen our understanding of quadratic functions.

To verify our zeros, we can plug them back into the original equation and see if we get y = 0. Let's start with x = 5:

y = (5)^2 - 12(5) + 35 = 25 - 60 + 35 = 0

And now for x = 7:

y = (7)^2 - 12(7) + 35 = 49 - 84 + 35 = 0

Both zeros check out! This gives us confidence that our solutions are correct. Now, let's think about the graph of the function. We know it's a parabola, and we know it intersects the x-axis at x = 5 and x = 7. Since the coefficient of the x^2 term (a) is positive (1 in our case), the parabola opens upwards. This means it has a minimum point, which is the vertex of the parabola.

The vertex lies on the axis of symmetry, which is exactly halfway between the zeros. In this case, the axis of symmetry is x = (5 + 7) / 2 = 6. To find the y-coordinate of the vertex, we plug x = 6 into the function:

y = (6)^2 - 12(6) + 35 = 36 - 72 + 35 = -1

So, the vertex is at the point (6, -1). Knowing the zeros and the vertex gives us a good idea of what the graph looks like. It's a U-shaped curve that crosses the x-axis at 5 and 7 and has its lowest point at (6, -1). Visualizing the graph helps solidify our understanding of the relationship between the zeros and the quadratic function itself. Graphing tools or software can be used to plot the function accurately and confirm our analysis. Verifying and visualizing are essential steps in problem-solving, especially in mathematics, to ensure accuracy and enhance comprehension.

Conclusion: Mastering Quadratic Zeros

Alright, guys, we've successfully found the zeros of the quadratic function y = x^2 - 12x + 35! We used two different methods – factoring and the quadratic formula – and arrived at the same solutions: 5 and 7. We also took the time to verify our answers and visualize the graph of the function. By understanding these methods and the concepts behind them, you've added valuable tools to your math toolkit.

Finding the zeros of a quadratic function is a fundamental skill in algebra, with applications in various fields, including physics, engineering, and computer science. Whether you're solving for projectile motion, designing structures, or modeling data, quadratic equations and their zeros often play a crucial role. The ability to confidently and accurately find these zeros is a testament to your mathematical understanding.

Remember, the key to mastering quadratic equations (and any math topic) is practice. The more you work through examples, the more comfortable and confident you'll become. Try solving different quadratic equations using both factoring and the quadratic formula to see which method works best for you in different situations. Don't be afraid to make mistakes – they're a natural part of the learning process. Each mistake is an opportunity to learn and improve.

So, keep practicing, keep exploring, and keep building your mathematical skills. You've got this! And who knows, maybe next time, you'll be the one explaining quadratic zeros to someone else. Keep up the great work, and happy solving! Now you have a solid foundation for tackling more complex problems involving quadratic functions and their zeros. Go forth and conquer those equations!