Quadratic Formula Calculator

Quadratic Formula Calculator:

This quadratic formula calculator solves quadratic equations, finds real and complex roots, and provides step-by-step solutions instantly.

Our free quadratic equation solver helps students, engineers, and professionals solve equations efficiently, improve understanding of concepts, and reduce errors from manual calculations.

What Is a Quadratic Equation?

A quadratic equation is a second-order polynomial equation that contains one variable, where the highest power of the variable is 2.

Standard Form and Key Components:

ax² + bx + c = 0

Where:

x is the variable
a, b, and c are real numbers
a ≠ 0, because if a = 0, the equation becomes linear

Key Concept:

The term quadratic comes from the Latin word quadratus, meaning “square,” which refers to the squared (²) term in the equation.

The solutions of a quadratic equation are called roots or zeros. These are the values of x that make the equation true.

Depending on the discriminant (b²−4ac), the roots can be:

Real and distinct
Real and equal (repeated)
Complex (non-real)

Nature of Roots:

Depending on the discriminant (b²−4ac), the roots can be:

1. Real and Distinct:

When the discriminant (b²−4ac > 0) is greater than zero. It means the curve of a parabola cuts the x-axis at two different points.

2. Real and Equal (Repeated):

When the discriminant (b²−4ac = 0) is equal to zero. In this case, the parabola touches the x-axis at only one point

3. Complex (Imaginary):

When the discriminant (b²−4ac < 0) is less than zero. In this case, the curve of a parabola does not cut the x-axis because the roots are not real

To quickly find the discriminant for given coefficients, try our discriminant calculator.

Real-World Applications:

Quadratic equations are commonly used in:

Projectile Motion (e.g., calculating the path of a ball)
Optimization Problems (finding maximum or minimum values)
Economics and Engineering (modeling relationships and systems

The Quadratic Formula – Explained:

The quadratic formula is used to find the solutions (roots) of a quadratic equation:

x = −b± √(b²−4ac) 2a

Where:

a is the coefficient of x²
b is the coefficient of x
c represents the constant term
b² - 4ac is known as the discriminant

How to Use the Quadratic Formula Calculator?

The calculator can be used in two ways: either by entering the coefficients (a, b, and c) of the equation directly or by providing the full quadratic equation.

Option #1: Entering Coefficients (a, b, c)

Input the values of the coefficients in their designated fields
Click on the “Calculate” button
See the discriminant (Δ), roots, and a graph, showing the visual of your quadratic equation

Option #2: Entering the Full Quadratic Equation

Enter the equation in the standard form
Press the “Calculate” button, and the calculator will automatically detect the coefficients a, b, and c
View the roots of the quadratic equation, discriminant (Δ), and the quadratic graph for a given equation to visualize the parabola curve

Example:

A ball is thrown upward from a height of 1.5 meters with an initial speed of 20 meters per second. Find the time it takes for the ball to hit the ground and solve using quadratic formula.

Solution:

Step #1: Write the equation of motion

The height of the ball at any time t (in seconds) is given by:

y = -4.9t² + 20t + 1.5

Here:

a = -4.9, b = 20, c = 1.5

The ball hits the ground when y = 0, so:

-4.9t² + 20t + 1.5 = 0

Step 2: Find the discriminant (Δ)

Δ = b² - 4ac
Δ = (20)² - 4(-4.9)(1.5)
Δ = 400 - (-29.4)
Δ = 429.4

Since the discriminant is positive (Δ > 0), there are two real roots.

Step 3: Apply the quadratic formula

t = (-b ± √Δ) (2a)

Substitute the values:

t = (-20 ± √429.4) (2 × -4.9)

t = -20 ± 20.72) -9.8

Step 4: Solve for both roots

First root:

t₁ = (-20 + 20.72) -9.8

t₁ = (0.72) -9.8

t₁ ≈ -0.07 seconds (not possible because time cannot be negative)

Second root:

t₂ = (-20 - 20.72) -9.8

t₂ = (-40.72) -9.8

t₂ ≈ 4.16 seconds

Step 5: Interpret the result

The ball reaches the ground after approximately 4.16 seconds.

The example demonstrates how the quadratic equation helps to describe the projectile motion. Here, the negative sign in front of the term t² indicates the downward acceleration because of the presence of the gravitational force. Solving this quadratic equation tells us about the time that the ball takes to return to the ground. For visualization, you can pair results with our quadratic graph calculator.

Key Takeaways:

The quadratic formula calculator online helps you quickly and easily solve quadratic equations. Whether you're checking homework, analyzing motion, or verifying solutions, this tool delivers accurate results with a clear, step-by-step breakdown.

Acts as a quadratic root calculator to solve equations quickly and accurately
Finds the discriminant as a quadratic discriminant calculator to determine the nature of roots
Helps visualize the relationship between coefficients and solutions
Ideal for students, teachers, and quick problem-solving

FAQ’s:

Can Quadratic Equations have no Real Solution?

Yes, the quadratic equation can have no real solution when the discriminant (b²−4ac) is less than zero. In this case, the roots are complex (imaginary), not real numbers.

What if the Discriminant is Negative?

When the discriminant is negative, then it means the quadratic equation has complex (imaginary) roots. It means the parabola does not intersect the x-axis.

Can this Calculator Solve Complex Roots?

Yes, this calculator can solve both real and complex (imaginary) roots.

Can I use the Calculator if a = 0?

No, you cannot. When a = 0, then the equation changes from a quadratic to a linear equation of the form bx + c = 0.

Does the calculator show Step-By-Step Work?

Yes, our quadratic formula calculator with steps not only provides the final result but also shows the step-by-step calculations, the discriminant (Δ), substitution into the quadratic formula, and calculated roots, so that you can understand how each value is derived.

How do you know if a Quadratic Equation has Two Solutions?

In simple terms, if discriminant Δ > 0, the parabola intersects the x-axis at two different points. It means the quadratic equation has two real solutions.

References:

Quadratic formula - Derivation by completing the square.
Solve Any Quadratic Equation - Understanding The Quadratic Formula.
By PurpleMath - Solve Each Quadratic Equation "The Quadratic Formula Explained".

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EN	DA	DE
TR	PL	ID
PT	FR	IT
ES	RU	FI
CS	JA	AR
KO