Solve quadratic equations instantly using the quadratic formula. Find real and complex roots with step-by-step solutions. Free online quadratic formula calculator with discriminant, graph, and detailed explanations.
ax² + bx + c = 0
Fractional values such as 3/4 can be used.
In algebra, a quadratic equation is any polynomial equation of the second degree with the following form:
Where x is an unknown, a is referred to as the quadratic coefficient, b the linear coefficient, and c the constant. The numerals a, b, and c are coefficients of the equation, and they represent known numbers. For example, a cannot be 0, or the equation would be linear rather than quadratic. A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing.
Let's back up for a second. A quadratic equation is any equation that looks like this:
Where a, b, and c are just numbers. The only rule is that a can't be zero (otherwise it's not quadratic anymore).
So 3x² + 2x - 5 = 0 is quadratic. x² - 9 = 0 is quadratic (b is just 0 here). Even -x² + 4x = 0 counts (c is 0).
The "quadratic" part comes from the Latin word "quadratus" which means "square." And that x² term? That's the square. That's what makes it quadratic.
Here it is. The formula that's saved millions of students' grades:
x = [-b ± √(b² - 4ac)] / 2a
Looks scary? It's not. Let's break it down piece by piece.
You've got three parts:
And that ± symbol? That means you get two answers. One with a plus, one with a minus. Most quadratic equations have two solutions.
Starting from ax² + bx + c = 0, divide by a and complete the square:
Using our calculator is stupidly simple. Here's what you do:
Let's try an example. Say you have x² + 5x + 6 = 0.
Here, a = 1, b = 5, c = 6. Type those in and you'll get x = -2 and x = -3. Check it: (-2)² + 5(-2) + 6 = 4 - 10 + 6 = 0. Works!
The most common mistake? Getting the signs wrong. If your equation is 2x² - 3x + 1 = 0, then b is -3, not 3. Type it in exactly as it appears.
See that part under the square root? b² - 4ac? That's called the discriminant. Fancy name, simple idea.
The discriminant tells you how many solutions your equation has. Think of it like a traffic light:
Here's a trick I learned: you can check the discriminant BEFORE you do the whole formula. If it's negative, save yourself the trouble and know there's no real answer.
Teachers love to say "you'll use this in real life." And honestly? They're right. Here's where quadratic equations show up:
Every time you throw a ball, shoot a basketball, or launch a rocket, the path is a parabola. The quadratic formula tells you when it'll hit the ground. For real. If you throw a ball from a height of 5 feet at 20 feet per second, the equation -16t² + 20t + 5 = 0 tells you it'll hit the ground after about 1.45 seconds.
Companies use quadratic equations to find the perfect price for their products. If profit = -2p² + 100p - 500 (where p is the price), the quadratic formula tells you the best price to charge. It's literally the difference between profit and loss.
Bridges, arches, and even some buildings use parabolic shapes. Engineers use quadratic equations to figure out the exact curve so the structure doesn't collapse. No pressure.
Game developers use quadratic equations for projectile motion, camera paths, and even some AI movement patterns. That perfect grenade toss in your favorite shooter? Quadratic formula.
After looking at hundreds of Reddit posts, here are the top mistakes students make:
The formula starts with -b. If b is 5, you get -5. If b is -3, you get 3. It flips the sign. Don't forget this.
The square root covers the entire b² - 4ac part. Not just b². Not just b² - 4. The whole thing. And the division by 2a happens after the square root.
If a = 0, it's not a quadratic equation. It's linear. The formula breaks. Check your equation first.
If your discriminant is negative, you'll get answers with "i" in them. That's the imaginary unit. It means √(-1). Don't panic - it's just math's way of saying "this doesn't have a real number answer." Your calculator handles it fine.
Ever wonder where the formula comes from? It's actually pretty cool. It comes from a method called "completing the square."
Here's the short version:
And that's how you get the formula. It's not magic - it's just algebra done carefully.
Here's the honest truth: factoring is faster when it works. But it doesn't always work. The quadratic formula always works.
Think of it like this: factoring is like taking the stairs - great if you're on the second floor. The quadratic formula is the elevator - it takes you anywhere, every time.
Use factoring when the numbers are small and you can see the factors. Use the quadratic formula for everything else.
Got an answer from the calculator? Here's how to check if it's right:
Plug your answer back into the original equation. If it equals zero (or close to it), you're good. Our calculator does this automatically, but it's a good habit to learn.
For example, if you got x = 2 for x² - 4 = 0, plug it in: 2² - 4 = 4 - 4 = 0. Works!
Our calculator works great on mobile. Here's what to watch out for:
Sometimes the calculator gives you answers that look strange. Here's what they mean:
People have been solving quadratic equations for over 4,000 years. The Babylonians had a method for it in 2000 BC. The formula as we know it today was finalized in the 17th century. That's older than calculus, older than Newton's laws, older than basically everything in modern math.
And here's something wild: there's a cubic formula (for equations with x³) and a quartic formula (for x⁴). But there's no general formula for equations with x⁵ or higher. Mathematicians proved it's impossible. So the quadratic formula is one of the last "simple" formulas before things get crazy.
The quadratic formula is x = [-b ± √(b² - 4ac)] / 2a. It solves any quadratic equation in the form ax² + bx + c = 0. You just plug in your numbers and get the answers. It works every single time, no matter how messy the equation looks.
First, make sure your equation is in the form ax² + bx + c = 0. Then identify a, b, and c. Plug them into the formula: x = [-b ± √(b² - 4ac)] / 2a. Calculate the discriminant (b² - 4ac) first, then take the square root, then do the rest. You'll get two answers from the ± sign.
The discriminant is b² - 4ac, the part under the square root. If it's positive, you get two real solutions. If it's zero, you get one real solution. If it's negative, you get two imaginary solutions (with "i" in them). It's like a quick preview of what kind of answers to expect.
The "i" stands for the imaginary unit, which is √(-1). It shows up when your discriminant is negative. This means the equation has no real number solutions. Don't worry - it's a real mathematical concept, and your calculator is handling it correctly.
The most common mistake is forgetting to flip the sign of b. The formula starts with -b, so if b is positive, you make it negative, and vice versa. Another big one is not putting the entire b² - 4ac under the square root. Always use parentheses if you're doing it by hand.
No - it only works for quadratic equations (ones with x² as the highest power). If your equation has x³ or higher, you need different methods. Also, a can't be zero, or it's not quadratic anymore. But for any ax² + bx + c = 0, the quadratic formula is your best bet.