Intro to Quadratic Equations - Online Tutor, Practice Problems & Exam Prep

Quadratic equations, or second-degree polynomials, take the form $a{x}^2+bx+c=0$. To solve them, factor the equation and set each factor to zero, yielding two possible solutions for x. Understanding coefficients a, b, and c is crucial, as they determine the equation's characteristics and solutions. Mastering this process is essential for solving quadratic equations effectively.

Introduction to Quadratic Equations

Hey, everyone. We've learned to solve a couple of different types of equations, and now we're going to add a new equation to the mix called a quadratic equation. Now a quadratic equation is going to be a bit more complicated than a linear equation, and it might even seem a bit overwhelming at times. But don't worry, I'm going to walk you through everything you need to know about quadratic equations in the next several videos starting with what a quadratic equation even is and then going into how to solve them. So let's go ahead and get started. Now if you take a linear equation, so like 2x-6=0, and you simply add an x squared term, so if I had a 3x2+2x-6=0, this is now a quadratic equation. A quadratic equation is also called a polynomial of degree 2 because the highest degree or power in my equation is this 2 on my x squared term. So whether you hear it called a quadratic equation or a second-degree polynomial, these two mean the same thing. You're often going to have to write quadratic equations in standard form. And the standard form of a quadratic equation is axx2+bx+c=0, where all of my terms are on the same side of my equation and they're all written in descending order of power. So my very first term, ax squared, has a power of 2, where my second term, bx, I can imagine an invisible one right here, so I have a power of 1. And then my last term is just a constant. There is no power. So I go from 2 to 1 to 0, descending order of power. You're not only going to have to write quadratic equations in standard form but you're also going to have to be able to identify each of those coefficients, a, b, and my constant c. Looking at my example up here, this 3x2+2x-6, if I asked you what a in that equation was, I would be able to say that this 3, since it's the coefficient of my x squared term, is a. Then b is the coefficient of my x term, which in this case is this positive 2. This is b. And then lastly, c, my constant term, is just the constant on the end of my equation. So in this case, it is a negative 6. I always want to make sure that I'm looking at that sign so that I can completely identify my constant or any other coefficient correctly. Let's go ahead and take a look at some other quadratic equations. I want to write each of these in a standard form and then identify what a, b, and c are.

Let's take a look at our first example. Here we have 5x2=x-3. Now to get this in standard form, I want all of my terms on the same side. So I'm going to go ahead and move this x over and then my negative 3 over so they're all on that left side. Now to get rid of the x on my right side, I need to go ahead and subtract it. Now remember, whatever you do to one side of the equation, you have to do to the other. That has not changed. And then I need to get rid of my 3 by adding it. So we'll, of course, cancel on that right side, leaving me on the left side with 5x2-x+3=0 because there is nothing left on that right side. Now I, of course, want these to be in descending order of power. So I have 2, and then my have my invisible one here, and then my 3 doesn't have a term, so I'm good. This is my quadratic equation in standard form, 5x2-x+3=0. Let's go ahead and identify a, b, and c in this equation. So a is going to be the coefficient of the first term of my x squared term. This is also called the leading coefficient because it's the very first coefficient in my quadratic equation. So a in this equation is 5. Then b is the coefficient of my x term. Here I have negative x, which I have an invisible one here, multiplying that x. So my b is negative 1. Then lastly, c, my constant term, is this positive 3 on the end here. So c here is 3, and that's all for that first example.

Let's go ahead and look at one more. Over here, I have -2x2+53=0. Now the first thing we want to do is get all of our terms to one side, and they already are here, so we're good. And then we want to check that they're in descending order of power. Well, my first term has a power of 2, and then my second term is just a constant. There is no power. So they actually already are written in descending order of power. And actually, this is already completely in standard form, so this is my quadratic equation in standard form. Let's go ahead and identify a, b, and c. Now a again is the coefficient of my x squared term. So in this case, I have a negative 2 multiplying that x squared, so that is a. B is the coefficient of my x term. But if I take a look at my equation, I actually don't have an x term at all. So b is actually going to be 0 because this would be like having a plus 0x stuck in that equation, which wouldn't do anything, which is why it's not there. And then c is my constant term, which in this case is actually a fraction, and it is 5 thirds. So it's totally fine for any of our a, b, and c to be a fraction or for b and c to even be 0. As long as I have that x squared term, it is still a quadratic equation. That's all for this video, and I'll see you in the next one.

Write the given quadratic equation in standard form. Identify a, b, and c. $-4x^2+x=8$

• A. a = - 4, b = 0, c = - 8
• B. a = - 4, b = 1, c = 8
• C. a = - 4, b = 1, c = - 8
• D. a = 2, b = 1, c = 0

Solving Quadratic Equations by Factoring

Hey everyone. Whenever we solve linear equations, we want to find a sum value for x that we can plug back into our original equation to make it true. And we want to do the same thing when solving quadratic equations. We want to find some value for x that makes our equation true. But whenever we solve linear equations, we were able to move some numbers around and then isolate x to get an answer. But what happens if I try to do the same thing with a quadratic equation? Well, looking at this x2-5x+4 over here, I might go ahead and move this 5x to the other side, maybe move my 4 as well leaving me with x2=5x-4. And then you might think, oh, we can just take the square root to get x by itself, which would leave me with x is equal to where do I go from there? What is the square root of 5x-4? So I can't actually solve a quadratic equation the same way I solved a linear equation. And more than that, there are actually often going to be 2 correct values for x when solving a quadratic equation. So how do we find those two values? Well, luckily, this is something else that we need is something that we already know how to do from the previous chapter, factoring. So let's go ahead and take a look at how we can factor to solve quadratic equations. Well, we're going to want to factor from standard form and then simply set each factor equal to 0. So if I have a quadratic equation like x2+x-6 is equal to 0 and then I go ahead and factor that into ( x+3) (x-2) equals 0, I can simply take each of those individual factors, so x+3 and x-2, set them equal to 0, and then solve for x. And the reason that I can do this is because these factors are multiplied together. So x+3⋅x-2 is going to equal 0 and anything times 0 is 0. So if either one of my factors, let's say that x+3 was 0, if I multiply that times anything, that would give me 0. So that might seem a little bit abstract right now. Let's go ahead and take a look at that in action. So in my example here, I want to solve this by factoring and I have x2-9x is equal to negative 20. So our first step here is going to be to write our equation in standard form. So I want to go ahead and get all my terms on the same side in descending order of power. So to do that, I just need to move my negative 20 to the other side, which I can do by adding 20 to both sides. It will cancel on that right side leaving me with x2-9x+20 is equal to 0 and we're done with step 1. Now step 2 is going to be to factor completely. Now remember, there are multiple different methods to factor, so let's go ahead and take a look at what method of factoring we should use for this equation. So we first want to look at how many terms this has. And it has this x2, negative 9x, x, and a positive twenty. So that is 3 terms, which tells me that I need to then say, does it fit a factoring formula? And looking quickly at my factoring formulas here, it doesn't seem to fit a factoring formula either. So my answer is no, which means that I then need to use the ac method to factor this quadratic. So let's go ahead and use the ac method here. So first, we want to look at a and c. So let's go ahead and make sure that we know what a, b, and c are here. So in this case, I have this kind of invisible one and that is my a, then b is negative 9 and c is 20. So if I multiply a and c together, a is just 1, so it's really just c. So ac is 20. So I want to find 2 factors that multiply to 20 and then add to b, which in this case is negative 9. So two factors that multiply to 20 add to negative 9. Since that b is negative, I know that both of my factors have to be negative. So let's think of factors of 20 that are negative. So I have negative 1 and negative 20, and then I have negative 2 and negative 10, and then I have negative 4 and negative 5. Now, negative 1 and negative 20 are going to add to negative 21, which is not negative 9. Negative 2 and negative 10 are going to add to negative 12, which is also not negative 9. But then I have negative 4 and negative 5, which do add to negative 9, so it turns out that negative 4 and negative 5 are my factors here. So once I have figured out what my factors are, I can go ahead and write this as (x-4)(x-5) is equal to 0, and step 2 is done. Now for step 3, I want to set my factors equal to 0 and then solve for x. So let's go ahead and look at each individual factor. So I have x-4. I'm setting that equal to 0, and then x-5 equals 0 as well. So let's go ahead and solve here. To solve for x with this x-4, I just need to add 4 to both sides leaving me with x is equal to 4. And then with x-5, I simply add 5 to both sides and I end up with x equals 5. And these are actually my solutions. I have completed step number 3. My solutions are x equals 4 and x equals 5. Now, because we wanted to find 2 values for x that we could plug back into our equation that makes it true, let's go ahead and make sure that that does happen. So let's first try this x equals 4 and go ahead and plug that in. So if I plug it into my original equation, I will get 4 squared minus 9 times 4 is equal to negative 20. Now 4 squared is 16, 9 times 4 is 36. So 16 minus 36 is equal to negative 20. 16 minus 36 is negative 20 and that's equal to negative 20. So negative 20 equals negative 20. That's definitely a true statement. I know that 4 is a value for x that makes it true. It's definitely one of my answers. Now we could do the same thing for 5 here but I'm going to leave that up to you. So that is all we need to do in order to solve quadratic equations by factoring. Let's go ahead and get some practice.

Solve the given quadratic equation by factoring. $3x^2+12x=0$

• A. $x=3,x=4$
• B. $x=0,x=-4$
• C. $x=-3,x=-4$
• D. $x=1,x=4$

Solve the given equation by factoring. $2x^2+7x+6=0$

• A. $x=3,x=\frac67$
• B. $x=-2,x=0$
• C. $x=2,x=\frac32$
• D. $x=-2,x=-\frac32$

Here’s what students ask on this topic:

What is a quadratic equation and how is it different from a linear equation?

A quadratic equation is a polynomial equation of degree 2, typically written in the form $a^{2}x+bx+c=0$. The key difference from a linear equation, which is of degree 1 (e.g., $\mathrm{ax}+b=0$), is that a quadratic equation includes an $x^2$ term. This makes the graph of a quadratic equation a parabola, whereas the graph of a linear equation is a straight line.

How do you write a quadratic equation in standard form?

To write a quadratic equation in standard form, you need to arrange it as $a^{2}x+bx+c=0$, where $a$, $b$, and $c$ are constants. Ensure all terms are on one side of the equation and ordered by descending power of $x$. For example, $5x^2-x+3=0$ is in standard form.

What are the coefficients in a quadratic equation and how do you identify them?

In a quadratic equation of the form $a^{2}x+bx+c=0$, the coefficients are $a$, $b$, and $c$. The coefficient $a$ is the number in front of $x^2$, $b$ is the number in front of $x$, and $c$ is the constant term. For example, in $3x^2+2x-6=0$, $a$ is 3, $b$ is 2, and $c$ is -6.

How do you solve a quadratic equation by factoring?

To solve a quadratic equation by factoring, follow these steps: 1) Write the equation in standard form $a^{2}x+bx+c=0$. 2) Factor the quadratic expression. For example, $x^2+x-6=0$ factors to $(x+3)(x-2)=0$. 3) Set each factor equal to zero: $x+3=0$ and $x-2=0$. 4) Solve for $x$: $x=-3$ and $x=2$. These are the solutions.

Why can a quadratic equation have two solutions?

A quadratic equation can have two solutions because it is a second-degree polynomial, meaning its graph is a parabola. When you solve the equation by factoring, you set each factor equal to zero, leading to two possible values for $x$. For example, in the equation $x^2-5x+6=0$, factoring gives $(x-2)(x-3)=0$. Setting each factor to zero, you get $x=2$ and $x=3$. These are the two solutions where the parabola intersects the x-axis.