Quadratic Equation Solver Calculator

What is a Quadratic Equation?

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:

• a = coefficient of x²
• b = coefficient of x
• c = constant (the number without x)

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.

How to Use the Quadratic Equation Calculator?

Here are the steps:

1. Enter Coefficient a: This is the number in front of x². Enter it in the first box.
2. Enter Coefficient b: This is the number in front of x. Enter it in the second box.
3. Enter Constant c: This is the plain number with no x. Enter it in the third box.
4. Click on Calculate Roots and you get your result with roots, discriminant, nature of roots and vertex point.

Suppose Rahul has a math exam tomorrow and his textbook gives him this equation: x² + 5x + 6 = 0

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:

• Root 1 (x₁) = -2
• Root 2 (x₂) = -3
• Discriminant = 1
• Nature = Real and Distinct

What is a Discriminant?

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.

What is the Nature of Roots?

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.

What is Vertex in a Quadratic Equation?

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.

How Does Our Quadratic Equation Calculator Work?

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.

Frequently Asked Questions