Quadratic Formula Calculator:
This quadratic formula calculator instantly finds the real and complex roots of any quadratic equation in the form ax² + bx + c = 0 (where a ≠ 0). By applying the quadratic formula, it provides accurate results for real, repeated, or imaginary roots, along with the discriminant value and a visual representation of the parabola. Our free quadratic equation solver helps students, engineers, and mathematicians solve equations more efficiently, understand the concept better, and reduce errors associated with manual calculations.
What Is a Quadratic Equation?
The quadratic equation is a second-order polynomial equation having a single variable.
Standard Form and Key Components:
ax² + bx + c = 0
Where:
- x is the variable
- a, b, and c are the real numbers
- a ≠ 0 (because if a = 0, the equation becomes linear)
The word quadratic comes from the Latin Word quadratus, meaning "squares." Meanwhile, the variable's highest power is a square. Remember, the solution of quadratic equations is known as their roots or zeros. The roots of a quadratic equation can be real, repeated, or complex numbers, depending on the discriminant. These roots are the values of “x” and make the quadratic equation true. To find the discriminant for the given coefficients, try our discriminant calculator online.
In real-world scenarios, the quadratic equations are often used to understand:
- Estimating a projectile’s trajectory
- In optimization problems, determining the maximum or minimum points
- To analyze economics, parabolic motion, or engineering systems
The Quadratic Formula – Explained:
The quadratic formula is:
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
The discriminant allows you to determine the nature of the roots:
- If b² - 4ac > 0, then it means the roots are real and distinct
- If b² - 4ac = 0, the roots are real and repeated
- If b² - 4ac < 0, then the roots are complex or imaginary
How the Quadratic Formula Is Derived?
The quadratic formula is derived by completing the square on a general quadratic equation. It's an easy way to solve for the value of “x” even when it's not possible to do the factoring.
Step #1: The general quadratic equation
ax² + bx + c = 0
Step #2: Divide the equation by ‘a’
Divide both sides of the equation by ‘a’ to make the coefficient of x² = 1.
x² + (b/a)x + (c/a) = 0
x² + bax + ca = 0
Step #3: Take the constant term to the right side of the equation
x² + bax = -ca
Step #4: Complete the square
- The half coefficient of x is b/a
- Its half is b/2a, now square it
- (b/(2a))² = b² / (4a²)
- Add the result on both sides of the equation:
x² + bax + b²4a² = -ca + b²4a²
Step #5: Have a perfect square of the left-hand side and write it
(x + b 2a )² = b² - 4ac 4a²
Step 6: Take the square root of both sides
x + b2a = ±√(b² - 4ac)2a
Step 7: Solve for x
Subtract b/(2a) from both sides of the equation
x = -b ± √(b² - 4ac)2a
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 graph a quadratic equation showing the parabola based on the provided inputs
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 complete the quadratic graph for a given equation to visualize the parabola curve
Types of Roots & What They Mean:
The value of the discriminant (Δ = b² - 4ac) defines the nature of the roots, whether they are real, repeated, or complex.
1. Real and distinct Roots (Discriminant > 0):
The nature of the roots of a quadratic equation depends on the value of the discriminant (Δ = b² - 4ac). If the discriminant is a positive value, then it means that the quadratic equation has two real and distinct roots. It means the curve of a parabola cuts the x-axis at two different points.
Example:
Equation: x² - 5x + 6 = 0
Here, Δ = (-5)² - 4(1)(6) = 25 - 24 = 1 (positive)
✅ Two real and distinct roots:
x₁ = 3, x₂ = 2
2. One Real Repeated Root (Discriminant = 0)
The quadratic equation has one repeated root when the value of the discriminant is equal to zero. It is also known as the double root. In this case, the parabola touches the x-axis at only one point.
Example:
Equation: x² - 4x + 4 = 0
Δ = (-4)² - 4(1)(4) = 16 - 16 = 0
✅ One real repeated root:
x = 2
Complex Roots (Discriminant < 0)
If you get a negative discriminant, then it means the quadratic equation has two complex roots that are conjugate to each other. In this case, the curve of a parabola does not cut the x-axis because the roots are not real.
Example:
Equation: x² + 4x + 5 = 0
Δ = (4)² - 4(1)(5) = 16 - 20 = -4 (negative)
✅ Two complex roots:
x = -2 ± i
Examples: Solve Real-Life Quadratic Problems
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.
FAQ’s:
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 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.
Key Takeaways:
Our quadratic formula calculator lets you easily and quickly solve quadratic equations. Whether you are verifying homework, analyzing the motion of an object, or verifying roots, this calculator provides accurate results with a detailed breakdown of each step.
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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