Need to solve a quadratic equation fast? Enter your values of a, b, and c. Our solve quadratic equation calculator gives you the roots, discriminant, nature of roots, and vertex in one click.
You have seen this equation before: ax² + bx + c = 0. This is a quadratic equation. It is one of the most common equations in math. Almost every student faces it in school and college exams.
The word “quadratic” comes from “quad” which means square. So any equation where the highest power of x is 2, that is a quadratic equation.
There are three parts you need to know:
One important rule. The value of “a” can never be zero. If a = 0, it is no longer a quadratic equation.
In simple words, A quadratic equation is a second-degree polynomial equation. Second degree means the highest power of the variable x is 2.
The fastest way to solve a quadratic equation is to use a calculator to solve quadratic equations. You skip the manual steps, avoid calculation mistakes, and get the full result including roots, discriminant, and vertex in seconds.
Here are the steps:
Suppose Rahul has a math exam tomorrow and his textbook gives him this equation: x² + 5x + 6 = 0. He needs to solve this quadratic equation and find the roots of the quadratic equation. Manually solving it takes time and there is a chance of mistakes. So Rahul opens our calculator.
He needs to find the value of x. Manually solving it takes time and there is a chance of mistakes. So Rahul opens our calculator. He enters a = 1, b = 5, c = 6. He clicks Calculate Roots.
He gets:
This is exactly what our solve quadratic equation calculator does. You get the same result for any equation in the same way, without writing a single step manually.
The discriminant is the part inside the square root: b² – 4ac. This value tells you what kind of roots your equation has before you even solve it.
There are mainly two outcomes you will see:
Discriminant is positive (> 0): Your equation has two real and different roots. This is the most common case in textbooks.
Discriminant is zero (= 0): Your equation has two equal real roots. Both roots will be the same number.
There is a third case too. When the discriminant is negative (< 0), the roots become complex or imaginary. That means there is no real solution. Our calculator handles this case as well and shows you the result in the form of a + bi.
Nature of roots means what type of answers your equation gives. It is directly connected to the discriminant value.
Our calculator shows you the nature automatically so you do not have to think about it separately.
The vertex is the turning point of the parabola formed by the quadratic equation. It is shown as (x, y).
The x coordinate of vertex = -b / 2a The y coordinate is calculated by putting that x value back into the equation.
Our calculator shows you the vertex automatically in the result section.
You enter your three values. The tool calculates the discriminant first. Then based on that discriminant value, it finds your roots. It also calculates the vertex point. All of this happens within a second after you click calculate.
Equation Preview: As you type your values, the equation preview at the top updates live. So you can see your equation forming in real time before calculating.
You will find quadratic equations in many places. Physics problems use it for projectile motion. Engineering uses it for structural design. Even finance uses it for profit and loss calculations.
There are two main ways to solve a quadratic equation: factoring and using the quadratic formula. Factoring works when the equation breaks down into clean integer roots. For example, x² + 5x + 6 = 0 factors neatly into (x + 2)(x + 3) = 0, giving roots of -2 and -3.
But not every equation factors cleanly. When the roots are decimals or complex numbers, the quadratic formula is the only reliable method. Our factoring quadratic equations calculator uses the formula approach so it works for every equation, not just the easy ones. You get your answer whether the roots are whole numbers, fractions, or complex.
If a = 0 then it is not a quadratic equation anymore. It becomes a linear equation. Our calculator will show you an alert if you enter zero as the value of a.
When discriminant is less than zero, the square root of a negative number is not real. In that case roots are written in the form a + bi where i is an imaginary unit.
Yes, There is no calculation limit. Just click the Reset button and enter new values. You can calculate as many times as you want.
Yes, We use the standard public quadratic formula which is used in every textbook and by every teacher worldwide.
Yes. For equations with clean integer roots, the result you get matches factoring. For equations with decimal or complex roots, the quadratic formula gives you the answer that factoring cannot.
Enter the values of a, b, and c in the calculator above and click Calculate Roots. You get both roots instantly along with the discriminant and the nature of the roots.
Yes. You enter the three coefficients, click calculate, and the tool solves the full quadratic equation for you using the standard quadratic formula.
Factoring rewrites the equation as a product of two expressions and finds roots by setting each to zero. Solving using the quadratic formula always works, even when the equation cannot be factored. Our calculator uses the formula method so no equation is left unsolvable.
Disclaimer: Our tool uses past data and trends, so it may be inaccurate. Always check with the official universities. If you find anything wrong, please report it immediately at [email protected], and we will update it as soon as possible.