Roots, Solutions, and Zeros
October 27, 2026

[00:00:05] So, let's start with the left one.
[00:00:08] In green, what do we call this thing?
[00:00:12] And this expression has a special name. What is it called?
[00:00:19] Quadratic polynomial. Let's just call it polynomial.
[00:00:26] Quadratic refers to a polynomial.
[00:00:32] So, we could just call it quadratic.
[00:00:40] And then the one in the middle, what do we call it?
[00:00:55] Sometimes we might say a quadratic equation in one variable.
[00:01:03] It's just a quadratic equation.
[00:01:07] And then the third one is what?
[00:01:12] Quadratic equation in two variables and we therefore answer y equals or f of x equals. So, we call it a quadratic function.
[00:01:22] They all factor.
[00:01:26] x plus 5 times x minus 3.
[00:01:34] Zero equals x plus five times x minus three. x plus 5, x minus 3.
[00:01:46] And, let me get my colors straight.
[00:01:53] f of x equals x plus 5 times x minus 3.
[00:02:00] The first line, they're all written in standard form, which is descending order.
[00:02:07] This line, they're all written in what form?
[00:02:12] Factored form. The leading coefficient for all of them is one.
[00:02:17] And from factor form, what can we tell real quick?
[00:02:23] This is very important.
[00:02:24] And the zeros are? Negative five and three.
[00:02:31] Okay, these same numbers here are called something else. What are they called?
[00:02:42] They're called solutions.
[00:02:48] What is a solution?
[00:02:52] A number, when plugged into the equation, makes the equation true.
[00:02:56] Yeah, so if it's an equation 1 variable, it's a number, then when replacing the variable makes the equation true.
[00:03:01] Okay, these same two numbers are not solutions here. Why are they not solutions?
[00:03:08] It's not an equation.
[00:03:12] Okay, and we call these same numbers roots.
[00:03:19] Okay.
[00:03:21] So they all sort of mean the same thing.
[00:03:25] But because we have different things, we have a polynomial or an expression.
[00:03:30] We have an equation in 1 variable, we have an equation in 2 variables, or a function. Functions have zeros.
[00:03:38] They also have solutions. Right, negative 5 is zero, but what's the actual solution?
[00:03:43] First off, it's an equation in 2 variables, so solutions are not numbers, they're ordered pairs.
[00:03:50] Zeros are x values that make y zero.
[00:03:52] Okay, so these do have solutions, they have an infinite number of solutions, but the two, where the y values are zero, we call them zeros.
[00:04:01] Okay, so new word we're learning today is a root.
[00:04:07] It's simply if this, it's values of x that make the polynomial, you plug them in equal to zero.
[00:04:16] Okay, or the value is zero. That's what a root is.
[00:04:32] Does this have roots, solutions, or zeros?
[00:04:36] Zero. It also has solutions, but in this context, we're talking about zeros, and the zeros are?
[00:04:46] Negative 4 and 3.
[00:04:48] You can put a comma, you can say and, it doesn't really matter. They're just numbers.
[00:04:55] Okay, and why do we have zeros?
[00:04:59] Because it's a function. Okay, it's x values that make f of x zero.
[00:05:05] Does this have roots, solutions, or zeros?
[00:05:10] And the roots are?
[00:05:12] Negative 5 and negative 2.
[00:05:18] Okay.
[00:05:20] What are we going to call 2 and negative 2 solutions?
[00:05:27] Or solutions.
[00:05:31] Okay. Equations in one variable have solutions.
[00:05:36] Especially when it's just two numbers.
[00:05:41] Okay, so that's the only difference.