Quadratic Formula Calculator | Solve ax² + bx + c = 0
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How It Works
Enter Coefficients
Input values for a, b, and c in ax² + bx + c = 0
Apply Formula
Calculate using x = (-b ± √(b² - 4ac)) / 2a
Calculation Table
| Quadratic Equation | Solutions | Type |
|---|---|---|
| x² - 5x + 6 = 0 | x = 2, 3 | Two Real |
| x² + 3x - 4 = 0 | x = 1, -4 | Two Real |
| 2x² - 7x + 3 = 0 | x = 3, 0.5 | Two Real |
| x² - 4 = 0 | x = ±2 | Two Real |
| x² - 6x + 9 = 0 | x = 3 | One Real |
| x² + 2x + 5 = 0 | x = -1 ± 2i | Complex |
| 3x² - 12x + 9 = 0 | x = 1, 3 | Two Real |
| 2x² + 5x - 3 = 0 | x = 0.5, -3 | Two Real |
Quadratic Formula
Common Examples
Two Real Solutions
x² - 5x + 6 = 0
Solutions: x = 2, 3
Discriminant > 0
One Real Solution
x² - 6x + 9 = 0
Solution: x = 3
Discriminant = 0
Complex Solutions
x² + 2x + 5 = 0
Solutions: x = -1 ± 2i
Discriminant < 0
Physics Application
-16t² + 64t + 80 = 0
Projectile motion timing
Real-world usage
What is a Quadratic Formula Calculator?
A quadratic formula calculator is a free online tool that solves quadratic equations for you. When you have an equation like ax² + bx + c = 0, this calculator finds the answers quickly. You just need to enter the three numbers (a, b, and c), and the calculator does all the math work.
The quadratic formula calculator uses a special math formula to find solutions. This formula works for any quadratic equation. It can find real solutions or complex solutions. The calculator shows you the exact answers in seconds.
Many students and professionals use this quadratic equation solver every day. It helps with homework, work projects, and real-life problems. The calculator is perfect when you need quick and accurate results.
Easy to Use
Just enter your three numbers (a, b, c) and get instant results. No complex math needed. The quadratic formula calculator does everything for you automatically.
Always Accurate
This quadratic equation solver uses the correct mathematical formula. It gives you precise answers every time. Perfect for students, teachers, and professionals.
Many Uses
Great for school math, physics problems, engineering work, and business calculations. Use it whenever you need to solve quadratic equations quickly.
Why Use Our Quadratic Formula Calculator?
Our free quadratic formula calculator makes solving equations simple and fast. You don't need to remember the complex formula or do difficult calculations by hand. Just enter your numbers and get the answer right away.
This quadratic equation solver is perfect for students learning algebra. It helps you check your homework answers. Teachers can use it to create examples for their classes. Engineers and scientists use it for their work projects.
Benefits for Students
- • Check homework answers quickly
- • Learn how quadratic equations work
- • Save time on math problems
- • Get help with test preparation
- • Understand different types of solutions
Benefits for Professionals
- • Solve engineering problems fast
- • Calculate projectile motion paths
- • Find optimal business solutions
- • Design curves and parabolas
- • Analyze data trends accurately
How to Use the Quadratic Formula Calculator
Using our quadratic formula calculator is very easy. You need to know the three numbers from your equation. Every quadratic equation looks like ax² + bx + c = 0. The letters a, b, and c are the numbers you need to enter.
Step-by-Step Instructions
Step 1: Find Your Numbers
Look at your equation and find a, b, and c:
- • a is the number before x²
- • b is the number before x
- • c is the number by itself
Example: 2x² + 5x - 3 = 0
a = 2, b = 5, c = -3
Step 2: Enter and Calculate
Put your numbers in the calculator:
- • Type 'a' in the first box
- • Type 'b' in the second box
- • Type 'c' in the third box
- • The answer appears automatically
Result: x = 0.5, -3
Two real solutions found
Important Tips
- • Remember to include negative signs if your numbers are negative
- • If there's no number before x², then a = 1
- • If there's no x term, then b = 0
- • If there's no constant term, then c = 0
- • The coefficient 'a' cannot be zero (that would make it a linear equation)
Understanding Quadratic Equation Solutions
When you use the quadratic formula calculator, you might get different types of answers. The calculator can find three types of solutions. Understanding these types helps you know what your answer means.
The type of solution depends on something called the discriminant. This is part of the quadratic formula. The discriminant tells us what kind of answer we will get before we finish the calculation.
Two Real Solutions
This happens when the discriminant is positive (greater than zero).
x² - 5x + 6 = 0
Solutions: x = 2, x = 3
The parabola crosses the x-axis twice
One Real Solution
This happens when the discriminant equals zero.
x² - 6x + 9 = 0
Solution: x = 3
The parabola touches the x-axis once
Complex Solutions
This happens when the discriminant is negative (less than zero).
x² + 2x + 5 = 0
Solutions: x = -1 ± 2i
The parabola doesn't touch the x-axis
What This Means in Real Life
Two Solutions: A ball thrown in the air hits the ground at two different times - when it goes up and when it comes down.
One Solution: A ball thrown perfectly reaches its highest point at exactly one moment.
Complex Solutions: Some math problems have answers that involve imaginary numbers. These are still valid mathematical solutions.
Business Use: Finding the break-even points for profit calculations often uses quadratic equations.
Real-World Uses of Quadratic Formula Calculator
The quadratic formula calculator is not just for school math. Many people use it in their daily work and life. Here are some real examples of when this quadratic equation solver becomes very helpful.
From sports to business, from science to art, quadratic equations appear everywhere. Our calculator makes it easy to solve these problems quickly and accurately.
Sports and Physics
Calculate when a basketball will reach the hoop
-16t² + 20t + 6 = 10
Find when ball reaches 10 feet high
Business and Profit
Find the best price to maximize profit
-2x² + 100x - 800 = 0
Find break-even points
Engineering Design
Design bridges and building curves
x² - 8x + 12 = 0
Find structural support points
Gardening and Farming
Calculate the best garden area for fencing
x² + 2x - 24 = 0
Find optimal rectangle dimensions
Computer Graphics
Create smooth curves in video games
3x² - 12x + 9 = 0
Calculate curve intersection points
Science Research
Analyze chemical reaction rates
2x² - 5x + 2 = 0
Find reaction equilibrium points
Why Professionals Choose Our Calculator
Fast Results: Get answers in seconds instead of spending minutes doing calculations by hand.
No Mistakes: Avoid human errors that can happen with manual calculations.
Always Available: Use it anytime, anywhere, on any device with internet access.
Free to Use: No cost, no registration, no limits on how many problems you can solve.
Easy to Understand: Clear results that show exactly what the solutions mean.
Reliable Formula: Uses the standard mathematical formula taught in schools worldwide.
Common Questions About Quadratic Formula Calculator
Many people have questions about using the quadratic formula calculator. Here are the most common questions with simple, easy-to-understand answers. These will help you use the calculator better and understand quadratic equations.
What is a quadratic formula calculator?
A quadratic formula calculator is a free online tool that solves equations like ax² + bx + c = 0. You enter three numbers (a, b, c) and it finds the solutions automatically. It uses the mathematical formula x = (-b ± √(b² - 4ac)) / 2a to get exact answers.
How do I use this quadratic equation solver?
First, look at your equation and find the numbers a, b, and c. Then type these numbers into the three boxes on the calculator. The answer appears right away. For example, if you have 2x² + 5x - 3 = 0, enter a=2, b=5, c=-3.
What types of answers can I get?
The quadratic formula calculator can give you three types of answers: two different real numbers, one repeated real number, or two complex numbers with imaginary parts. The type depends on your equation's discriminant (b² - 4ac).
What if my 'a' coefficient is zero?
If a = 0, your equation is not quadratic anymore - it becomes linear. The quadratic formula calculator won't work because you need a² term. Instead, solve bx + c = 0 by using x = -c/b.
Can I use negative numbers in the calculator?
Yes! The quadratic equation solver works with positive and negative numbers. Just make sure to include the negative sign when you type the number. For example, if your equation is x² - 4x + 3 = 0, enter a=1, b=-4, c=3.
What are complex solutions and when do they happen?
Complex solutions have imaginary numbers (with 'i'). They happen when the discriminant (b² - 4ac) is negative. For example, x² + 2x + 5 = 0 gives x = -1 ± 2i. These are still valid mathematical answers, just not real numbers.
Is this quadratic formula calculator accurate?
Yes, our calculator is completely accurate. It uses the exact mathematical formula taught in schools and universities. The results are precise and reliable for homework, professional work, and research projects.
When should I use the quadratic formula instead of factoring?
Use the quadratic formula calculator when factoring is hard or impossible. It works for any quadratic equation, even when the numbers don't factor nicely. It's also faster when you need exact decimal answers instead of fractions.
Can I check my homework answers with this calculator?
Absolutely! This quadratic equation solver is perfect for checking homework. Enter your equation's coefficients and compare the results with your manual calculations. It helps you learn and catch any mistakes you might have made.
Do I need to pay to use this quadratic formula calculator?
No, this calculator is completely free! You don't need to register, download anything, or pay any fees. Use it as many times as you want for school, work, or personal projects. It works on computers, tablets, and phones.