Find The Roots Of The Equation Calculator

Find the Roots (ax² + bx + c = 0)

Enter the coefficients a, b, and c for the quadratic equation ax² + bx + c = 0 to find its roots.

What are the Roots of a Quadratic Equation?

The Roots of a Quadratic Equation (ax² + bx + c = 0) are the values of x for which the equation holds true. These roots are the points where the graph of the quadratic function y = ax² + bx + c intersects the x-axis. Finding the Roots of a Quadratic Equation is a fundamental concept in algebra.

Anyone studying algebra, or professionals in fields like physics, engineering, and finance who work with quadratic models, should understand how to find the Roots of a Quadratic Equation. Common misconceptions include thinking every quadratic equation has two distinct real roots; sometimes they have one repeated real root or two complex roots. Our Roots of a Quadratic Equation calculator helps clarify this.

Roots of a Quadratic Equation Formula and Mathematical Explanation

The most common method to find the Roots of a Quadratic Equation is using the quadratic formula:

x = [-b ± √(b² – 4ac)] / 2a

The term inside the square root, Δ = b² – 4ac, is called the discriminant. The discriminant tells us about the nature of the roots:

• If Δ > 0, there are two distinct real roots.
• If Δ = 0, there is exactly one real root (or two equal real roots).
• If Δ < 0, there are two complex conjugate roots.

If a = 0, the equation becomes linear (bx + c = 0), and the root is simply x = -c/b, provided b ≠ 0.

Variables Explained

Variable	Meaning	Unit	Typical Range
a	Coefficient of x²	Dimensionless	Any real number (a ≠ 0 for quadratic)
b	Coefficient of x	Dimensionless	Any real number
c	Constant term	Dimensionless	Any real number
Δ	Discriminant (b² – 4ac)	Dimensionless	Any real number
x₁, x₂	Roots of the equation	Dimensionless	Real or Complex numbers

Variables used in the quadratic formula for finding the Roots of a Quadratic Equation.

Practical Examples (Real-World Use Cases)

Example 1: Projectile Motion

The height `h` of an object thrown upwards can be modeled by h(t) = -16t² + vt + h₀, where `t` is time, `v` is initial velocity, and `h₀` is initial height. To find when the object hits the ground (h=0), we solve -16t² + vt + h₀ = 0. If v=48 ft/s and h₀=0, we solve -16t² + 48t = 0. Here a=-16, b=48, c=0. The Roots of a Quadratic Equation are t=0 (start) and t=3 seconds (hits ground).

Example 2: Area Problem

Suppose you have a rectangular garden with length 5 meters longer than its width, and the area is 36 m². If width is `w`, length is `w+5`, so area A = w(w+5) = w² + 5w = 36. We solve w² + 5w – 36 = 0. Here a=1, b=5, c=-36. Using the formula, the Roots of a Quadratic Equation are w=4 and w=-9. Since width must be positive, w=4 meters.

How to Use This Roots of a Quadratic Equation Calculator

1. Enter Coefficient 'a': Input the number multiplying x² in the 'a' field. It cannot be zero for a quadratic equation.
2. Enter Coefficient 'b': Input the number multiplying x in the 'b' field.
3. Enter Coefficient 'c': Input the constant term in the 'c' field.
4. Calculate: Click "Calculate Roots" or observe the results updating as you type.
5. Read Results: The calculator displays the discriminant, the nature of the roots, and the values of root 1 (x₁) and root 2 (x₂). It will specify if the roots are real or complex.
6. View Graph: The graph shows the parabola y=ax²+bx+c and where it intersects the x-axis (if roots are real).

This Roots of a Quadratic Equation calculator gives you instant results, helping you understand the solutions to your quadratic equations quickly.

Key Factors That Affect Roots of a Quadratic Equation Results

1. Value of 'a': Affects the width and direction of the parabola. If 'a' is close to zero, the parabola is wide. If 'a' is large, it's narrow. If 'a' is 0, it's not a quadratic equation.
2. Value of 'b': Influences the position of the axis of symmetry of the parabola (x = -b/2a).
3. Value of 'c': Represents the y-intercept of the parabola (where x=0).
4. The Discriminant (b² – 4ac): The most crucial factor determining the nature of the Roots of a Quadratic Equation – whether they are two distinct real, one real, or two complex roots.
5. Sign of 'a': If 'a' > 0, the parabola opens upwards; if 'a' < 0, it opens downwards. This affects whether the vertex is a minimum or maximum.
6. Ratio b²/4a vs c: The comparison between b²/(4a) and c is directly related to the discriminant and the number and type of roots for the Roots of a Quadratic Equation.

Frequently Asked Questions (FAQ)

What is a quadratic equation?

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term that is squared. The standard form is ax² + bx + c = 0, where a, b, and c are coefficients and a ≠ 0.

What are the roots of a quadratic equation?

The roots (or solutions) of a quadratic equation are the values of x that satisfy the equation – where the graph y = ax² + bx + c crosses the x-axis. Our Roots of a Quadratic Equation calculator finds these values.

What is the discriminant?

The discriminant is the part of the quadratic formula under the square root sign: Δ = b² – 4ac. It determines the number and type of the Roots of a Quadratic Equation.

Can a quadratic equation have no real roots?

Yes, if the discriminant is negative (Δ < 0), the quadratic equation has no real roots. It will have two complex conjugate roots. The Roots of a Quadratic Equation calculator identifies this.

Can a quadratic equation have one root?

Yes, if the discriminant is zero (Δ = 0), the quadratic equation has exactly one real root (a repeated root). The vertex of the parabola touches the x-axis.

What if 'a' is 0?

If 'a' is 0, the equation becomes bx + c = 0, which is a linear equation, not quadratic. It has one root x = -c/b (if b ≠ 0).

How do I find the roots if the equation is not in the form ax² + bx + c = 0?

You first need to rearrange the equation algebraically into the standard form ax² + bx + c = 0 to identify a, b, and c before using the Roots of a Quadratic Equation formula or calculator.

Are the roots always numbers?

The roots are always numbers, but they can be real numbers or complex numbers (involving 'i', the square root of -1).