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Welcome to Algebra I: Finding Zeros of Quadratic Functions

13 minutes

Hey, guys. Welcome to Algebra 1. Today we're going to focus on finding zeroes of quadratic functions. Your knowledge of factoring and the coordinate plane and x-intercepts will come in handy during this lesson. You ready to get started? Let's go.

(female narrator) Example 1: Find the zeroes of a quadradic function. y equals x squared plus x minus 20. Taking a look at this first example, you're asked to find the zeroes of this quadratic function. Just keep in mind that when finding a zero, you're being asked to find an x-intercept. Here you're actually asked to find the zeroes, plural, because typically in our quadratic functions, you have two x-intercepts. There will be some times where you don't, but for the most part, with problems in this lesson, we're going to have two zeroes. We know, to find an x-intercept, that y is always 0, right? X-intercepts always take the form of x is some numerical value, but y is 0. When given this quadratic function, in order to find the x-intercepts, we're going to want to start by replacing y with 0. So, 0 equals x squared plus x minus 20. Okay? Now we have this quadratic equation. Knowing what you do about solving quadratic equations, you know that to solve this, we start by factoring the right side. We'll need to find two numbers that multiply to give us negative 20, but combine to give us positive 1. For a refresher on factoring, check out that video. That will take you through the steps to factor quadratics. For this lesson, I'll assume you know the ins and outs. We'll just dive in and get this right side factored. The pair of numbers with a product of negative 20 but a sum of positive 1 are positive 5 and negative 4, because 5 times negative 4 is negative 20, and 5 plus negative 4 is positive 1. Okay? I'm going to rewrite that right side by a product of its factors: x plus 5 and x minus 4. Now to get the zeroes, or to get the x-intercepts, I need to solve these two mini equations. I'll set both parts equal to 0, individually, and see what value of x in this first group would give this whole group a value of 0, and what value of x would give the second group a value of 0. Here we go: x plus 5 equals 0. I'm going to solve that. And I'll solve x minus 4 equals 0. To solve the first one, subtract 5 from both sides, so x equals negative 5. To solve the second one, add 4 to each side, so x equals positive 4. I can tell on this quadratic function that my zeroes are located at negative 5 and 4, or if I wanted to write them as ordered pairs,

(negative 5, 0) and (4, 0). There you go-- you've got your zeroes for this quadratic function. Okay? Let's try another one. Here we're asked to find the zeroes of the quadratic function, y equals x squared minus 3x minus 10. Okay? Again, when you're asked to find the zeroes, you're really being asked to find the x-intercepts. Those are interchangeable terms. For my x-intercept, I always like to write myself a little hint that the x value is some numerical value, but y must be 0. To find the zeroes of this function, I'm going to start out by replacing that y with a 0. So, I'll have 0 equals x squared minus 3x minus 10. Then I'm going to get this right side factored, to solve this quadratic. I'm looking for two numbers that have a product of negative 10, but when added together, they get negative 3. For this case it's going to be negative 5 and positive 2. Because negative 5 times positive 2 is negative 10, and negative 5 plus positive 2-- or negative 5 plus 2-- is negative 3. Okay? Let's rewrite that right side:

(x minus 5) and (x plus 2). And then let's solve these two-- what I like to call mini equations-- two one-step equations, to get exactly the value of each of those x-intercepts. We'll have x minus 5 equals 0, and we'll have x plus 2 equals 0. So we'll add 5 to each side here. Okay? So, that'll wipe out, And x equals 5 is one of my zeroes. Then I'll subtract 2, subtract 2 over here. And x equals negative 2 is my other zero. Now if I want, I could stop here. This is an acceptable answer. My zeroes are at x equals 5 and negative 2. Or I could also write these as ordered pairs. I could represent them as (5, 0) and-- let's get a little more room here--

(negative 2, 0). Either way is fine. Okay. All right? I do believe it's time for you to try one. Press pause. Take a few minutes. And find the zeroes of this quadratic function. To compare your answers against mine, press play.

(female narrator) Find the zeroes of the quadratic function: y equals x squared minus 7x plus 12. All right, let's see how you did. First, if I'm being asked to find the zeroes, I know I'm being asked to find the x-intercepts. And I know those always take the form of x as some number, y is 0. So to find the zeroes of this quadratic, I'll start out by replacing y with 0. So 0 equals x squared minus 7x plus 12. Now I have a quadratic equation, and in order to solve it, what I'm going to do is get that right side factored. I need two numbers that have a product of positive 12 but a sum of negative 7. Going through the factors, you'll find that it is negative 4 and negative 3. Because negative 4 times negative 3 is positive 12, and negative 4 plus negative 3 is negative 7. Let's rewrite this right side as a product of factors.

(x minus 4), (x minus 3). Okay? Let's get a little more workspace here. Now it's time to solve what I like to call the mini equations, so we can see what the value of these x-intercepts are. I'll set x minus 4 equal to 0, and I'll set x minus 3 equal to 0. Okay? To solve this one, plus 4 plus 4, so x equals 4. To solve the other, plus 3 plus 3, x equals 3. So there are your x-intercepts. You have one at 4 and one at 3. Or, if you wrote them as ordered pairs, you have one located at (4, 0), and another located at (3, 0). All right? Good job on that! Now let's take a look at the graphs of some quadratic functions, to make sure that you understand that connection between finding the zeroes algebraically, when you're given an algebraic representation of a quadratic function, as well as finding the zeroes when you're given a graphical representation of a quadratic function. Now let's get this locked in place so it doesn't move on us. Again, if you're being asked to find the zeroes, what are you actually being asked to find? You're really being asked to find those x-intercepts. In the case of a graph, just zone right in to the x-axis and see where your parabola crosses the x-axis. Here we have two places really close to each other. We have one here and we have another right there. So we have one zero located at (negative 2, 0). I'm going to write that as an ordered pair. And then we have another zero located at (negative 3, 0). There are our zeroes for this quadratic function. Okay? Let's try another one. Let me get this locked in place first. Switch to the eraser-- there we go. Now let's get this locked in place so it doesn't move on us when we start analyzing it. We're being asked to find our zeroes. We're actually being asked to find the x-intercepts, Zone right in to the x-axis and see where your parabola crosses it. You have two places here-- I've got the highlighter. We have one zero right here-- there's one x-intercept-- and another right there. If I wrote them as ordered pairs, I'd say I have one zero at (negative 1, 0)-- that's the location of that point-- and then I have another zero located at 1, 2, 3, 4...0, because that's the location of that one. There are my two zeroes for this quadratic function. All right? Now you give this one a try. Press pause, take a minute, and locate the zeroes of this quadratic function. To compare your answer against mine, press play.

(female narrator) The bottom of the parabola is just under the x-axis and crosses the x-axis at 1 and 2. All right? Let's see how you did. Let me get my graph locked in place here so it doesn't start to move on me. Let's locate these zeroes. If I'm asked to find the zeroes, I'm really finding the x-intercepts. And I have one located right here, and another right there. I'm just going to write those locations as ordered pairs. So, I have (1, 0) right here at 1, 0, believe it or not. And I have another zero located right here at (2, 0). There are the locations of my two zeroes for my quadratic function. All right, great job, guys, finding the zeroes of your quadratic functions. I hope you saw how factoring and your knowledge of the coordinate plane came in handy for this lesson. Hope to see you back here soon for more Algebra 1. Bye!

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In this program, students learn how to find the zeros of quadratic functions. These points occur where the graph of the quadratic equation crosses the x-axis. Part of the "Welcome to Algebra I" series.

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Runtime: 13 minutes

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