The Quadratic Formula - Online Tutor, Practice Problems & Exam Prep

The quadratic formula, given by $-b\pm \sqrt{b^2-4*a*c}$ divided by 2*a, allows for solving any quadratic equation. The discriminant, $b²-4*a*c$, indicates the nature of solutions: positive for two real solutions, zero for one real solution, and negative for two imaginary solutions. Understanding these concepts is crucial for efficiently solving quadratic equations.

Solving Quadratic Equations Using The Quadratic Formula

Hey everyone. You just finished learning 3 different methods of solving quadratic equations, and you might be wondering how there's anything left and why we have yet another method to learn. But the quadratic formula that we're going to talk about now is really great because it's going to work for any quadratic equation. So even if you were to forget every single other method, as long as you remember the quadratic formula, you're going to be able to solve any quadratic equation that gets thrown at you. So let's go ahead and jump in.

The quadratic formula is based on the standard form of a quadratic equation ax2+bx+c, and we're going to use a, b, and c in order to compute our solutions. So the quadratic formula is −b±b2−4ac2a. Now that formula might look a little bit complicated right now and, unfortunately, it is something that you're going to have to memorize. But a way that I was able to memorize the quadratic formula was by using the quadratic formula song. Now, I'm not going to sing it for you right now, but it's going to be something that you should search up on your own to commit this formula to memory.

So when do we want to use the quadratic formula? Well, like I said, you can use the quadratic formula whenever you want, and some clues that you might want to use it are that you can't easily factor or you're just otherwise unsure what method to use. Let's go ahead and take a look at an example here. So I have x2+2x−3 is equal to 0. Let's go ahead and take a look at our first step, which is to write our equation in standard form. Now it looks like my equation is already in standard form, all of my terms are on the same side in the in descending order of power, so I can go ahead and move on to step 2.

Now step 2 is simply to just plug everything into my quadratic formula, so let's go ahead and do that here. I'm first going to label a, b, and c in my equation so that it's easy for me to just take them and plug them in. So in this case, I have an invisible one in front of that x squared and that is my a, b is this positive 2, and then c is negative 3. Make sure that you're paying attention to the signs here. So plugging this in, I get −2±162. Now that we've plugged everything in, we've completed step 2, and all that we have left to do from here is algebra.

We're just going to compute and simplify our solutions now. So let's start with what's in our radical here and just simplify that. Now I know that the square root of 16 is just 4, so this becomes −2±42. Now is when we're going to want to split our solution into the plus and the minus. So let's go ahead and do that. This splits into −2+42 and −2−42. And now I can just simplify those solutions down. So −2 + 4 is going to give me positive 2 over 2, which is just equal to 1. And then −2 − 4 is going to give me −6 divided by 2, which is going to give me −3. And I'm done. These are my solutions. So for this quadratic equation, my solutions are x equals 1 and x equals −3 and I'm completely done there.

I know that the quadratic formula can look a little intimidating at first but it really just comes down to the algebra once you plug everything in. Let's go ahead and take a look at one more example. So here I have x2−5x is equal to −1. Let's go ahead and start back at step 1, which is to write our equation in standard form. Now here, it looks like I need to go ahead and move my −1 over. So I can do that by simply adding 1 to both sides. And so this will become x2−5x+1 is equal to 0, because that one canceled on that right side. So step 1 is good. That's in standard form now. We can move on to step 2 and just plug everything in. So again, I'm going to go ahead and label a, b, and c here. I again have an invisible one for that a, b is negative 5, and c is positive 1.

So let's go ahead and complete step 2 and plug everything in. So I have −−5±212. Okay. So step 2 is done. Now we're just left to simplify. Just do all of that algebra. So let's first start with everything under that radical. Now I can't simplify this anymore. Even if I split it, nothing would simplify because I just have that square root of 21 can't go anywhere from there. So my solutions are just 5 plus or minus the square root of 21 over 2. Now I can of course split these into their plus and minus just to show both answers separately and I would just write 5 + 21 over 2 and 5 − 21 over 2. Now, as you saw with some of our other methods, you're going to have different types of answers here. So you might have whole numbers, you might also have some combination of fractions and radicals, and either is fine. That's all there is to the quadratic formula let me know if you have any questions.

Solve the given quadratic equation using the quadratic formula. $3x^2+4x+1=0$

• A. $x=3,x=-1$
• B. $x=-\frac13,x=-1$
• C. $x=-3,x=-1$
• D. $x=\frac13,x=-1$

Solve the given quadratic equation using the quadratic formula. $2x^2-3x=-3$

• A. $x=\frac34+\frac{i\sqrt{15}}{4},x=\frac34-\frac{i\sqrt{15}}{4}$
• B. $x=\frac34+\frac{5i}{4},x=\frac34-\frac{5i}{4}$
• C. $x=\frac{3+\sqrt{15}}{4},x=\frac{3-\sqrt{15}}{4}$
• D. $x=3+i\sqrt{15},x=3-i\sqrt{15}$

The Discriminant

Hey everyone. So we've just finished learning about all of the different methods of solving quadratic equations, but we're going to take a quick detour here to talk about the discriminant. Now the discriminant might feel a little bit random here, and you might be wondering why you need to learn about it, but the reason I'm telling you about it at all is because you're going to be asked a very specific type of question about how many real or imaginary solutions a quadratic equation has, and you're not going to want to have to go through fully solving it just to answer that it has one real solution. So that's where the discriminant comes in. So I'm going to show you exactly how to use the discriminant to answer this type of question really quickly. Let's go ahead and dive in.

So the discriminant is just the expression underneath the radical in the quadratic formula. So it's just b2 - 4 × a × c . And the sign of the discriminant is going to tell us the number and the type of solutions. So if the discriminant is positive, that means that my quadratic equation is going to have 2 real solutions. If it's 0, that tells me that I only have one real solution. And if it's negative, that tells me that I actually have no real solutions, and I have 2 imaginary ones. So whether it's positive, 0, or negative, I can see easily and quickly whether it has 2 real solutions, one real solution, or none at all.

So let's go ahead and look at a couple of different quadratic equations and identify the number and the type of solutions they have. So looking at my first one here, I have 2x2+3x-2=0. Now if I want to calculate my discriminant here, let's go ahead and label a, b, and c to make it a bit easier to plug in. So a here is 2, b is 3, c is negative 2. Plugging that all into my discriminant formula, I have 32-4×2×(-2). Now simplifying that, 32+16 is 9 + 16, which gives me 25. Now this is a positive number, so that tells me that I have 2 real solutions. So this quadratic equation has 2 real solutions.

Let's look at another example. So I have 4x2+x+2=0. Let's go ahead and label a, b, and c. Here, a is 4, b is this 1, I have an invisible one there, and c is 2. Now calculating the discriminant, I have 12-4×4×2. So simplifying this, 1 minus 32 is going to give me negative 31. So I am left with a negative number, which tells me that I have no real solutions and I have 2 imaginary ones. So here I have 0 real and 2 imaginary solutions.

Let's take a look at one final example here. So I have x2-10x+25=0. Now going ahead and labeling a, b, and c, a here is 1, b is negative 10. Make sure to look at that sign, and c is positive 25. So plugging this into my discriminant formula, I have -102-4×1×25. Now simplifying this, 100 minus 100 is going to give me 0. So finally here, I have that my discriminant is 0, which is the last option that I have for my discriminant. And that tells me that I only have one real solution, And that's all there is to the discriminant. Now whenever you're asked this question, you can do it really quickly. Let me know if you have any questions.

Determine the number and type of solutions of the given quadratic equation. Do not solve. 

$x^2+8x+16=0$

• A. 2 real solutions
• B. 1 real solution
• C. 2 imaginary solutions

Determine the number and type of solutions of the given quadratic equation. Do not solve. 

$-4x^2+4x+5=0$

• A. 2 real solutions
• B. 1 real solution
• C. 2 imaginary solutions

Here’s what students ask on this topic:

What is the quadratic formula and how is it used to solve quadratic equations?

The quadratic formula is used to solve quadratic equations of the form $a^2+bx+c=\; 0$

. The formula is given by:

$\frac{-b\pm \sqrt{b^2}}{2a}$

To use the quadratic formula, identify the coefficients $a$, $b$, and $c$ from the equation, plug them into the formula, and simplify to find the solutions for $x$. This method works for any quadratic equation, even when factoring is difficult or impossible.

What is the discriminant in the quadratic formula and what does it tell us about the solutions?

The discriminant in the quadratic formula is the expression under the square root: $b^2$. It indicates the nature of the solutions:

• If the discriminant is positive, there are two real solutions.
• If the discriminant is zero, there is one real solution.
• If the discriminant is negative, there are two imaginary solutions.

By calculating the discriminant, you can quickly determine the number and type of solutions without fully solving the quadratic equation.

How do you solve a quadratic equation using the quadratic formula?

To solve a quadratic equation using the quadratic formula, follow these steps:

1. Write the equation in standard form: $a^2+bx+c=\; 0$
2. Identify the coefficients $a$, $b$, and $c$.
3. Plug the coefficients into the quadratic formula: $\frac{-b\pm \sqrt{b^2}}{2a}$
4. Simplify the expression under the square root (the discriminant).
5. Calculate the two possible values for $x$ by using the plus and minus signs.

This will give you the solutions to the quadratic equation.

When should you use the quadratic formula instead of other methods?

The quadratic formula can be used to solve any quadratic equation, making it a versatile tool. It is especially useful when:

• The equation cannot be easily factored.
• You are unsure which method to use.
• You need to find the exact solutions, including complex numbers.

While other methods like factoring, completing the square, or graphing can be quicker for certain equations, the quadratic formula is a reliable method that works in all cases.

Can the quadratic formula be used to solve equations with complex solutions?

Yes, the quadratic formula can be used to solve equations with complex solutions. When the discriminant $b^2$ is negative, the solutions will be complex numbers. The formula will yield solutions in the form of $-b\pm \sqrt{b^2}$, where the square root of a negative number introduces the imaginary unit $i$. These solutions are written as $-b\pm i\sqrt{-b^2}$.