Find Roots Of Equation Calculator

Easily find the roots of any quadratic equation using our Roots of Equation Calculator. Input a, b, and c to get the solutions.

Enter Coefficients

Summary Table

Parameter	Value
Coefficient a	1
Coefficient b	-3
Coefficient c	2
Discriminant Δ	–
Root x1	–
Root x2	–
Nature of Roots	–

Summary of input coefficients and calculated roots.

What is a Roots of Equation Calculator?

A Roots of Equation Calculator, specifically for quadratic equations (of the form ax² + bx + c = 0), is a tool used to find the values of 'x' that satisfy the equation. These values are called the "roots" or "solutions" or "zeros" of the equation. Essentially, they are the points where the graph of the quadratic function y = ax² + bx + c intersects the x-axis. Our Roots of Equation Calculator quickly determines these roots based on the coefficients a, b, and c you provide.

Anyone studying algebra, engineering, physics, economics, or any field that uses quadratic models can benefit from a Roots of Equation Calculator. It saves time and helps verify manual calculations. Common misconceptions include thinking all equations have real roots (they can be complex) or that there's always only one root (quadratics usually have two).

Roots of Equation Calculator: Formula and Mathematical Explanation

The roots of a quadratic equation ax² + bx + c = 0 (where a ≠ 0) are found using the quadratic formula:

x = [-b ± √(b² – 4ac)] / 2a

The expression inside the square root, Δ = b² – 4ac, is called the discriminant. The value of 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.

Variables Table

Variable	Meaning	Unit	Typical Range
a	Coefficient of x²	Dimensionless	Any real number except 0
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

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 + v₀t + h₀, where t is time, v₀ is initial velocity, and h₀ is initial height. To find when the object hits the ground (h=0), we solve -16t² + v₀t + h₀ = 0. If v₀=64 ft/s and h₀=0, we solve -16t² + 64t = 0. Using the Roots of Equation Calculator with a=-16, b=64, c=0, we find t=0 and t=4 seconds. The object is at ground level at t=0s and t=4s.

Example 2: Area Problem

A rectangular garden has a length 5 meters more than its width, and its area is 36 square meters. If width is 'w', length is 'w+5', and area is w(w+5) = 36, so w² + 5w – 36 = 0. Using the Roots of Equation Calculator with a=1, b=5, c=-36, we get w=4 and w=-9. Since width cannot be negative, the width is 4 meters.

How to Use This Roots of Equation Calculator

1. Enter Coefficient a: Input the value of 'a' (the number multiplying x²) into the first field. Remember, 'a' cannot be zero.
2. Enter Coefficient b: Input the value of 'b' (the number multiplying x) into the second field.
3. Enter Coefficient c: Input the constant term 'c' into the third field.
4. Calculate: Click the "Calculate Roots" button, or the results will update automatically as you type.
5. Read Results: The calculator will display the discriminant, the nature of the roots, and the values of the roots (x1 and x2). If the roots are complex, they will be shown in the form a + bi and a – bi. The graph will also update.
6. Interpret Graph: The graph shows the parabola y = ax² + bx + c. The points where it crosses the x-axis are the real roots. If it doesn't cross, the roots are complex.

The Roots of Equation Calculator gives you the solutions directly. If the roots are real, these are the x-intercepts of the parabola.

Key Factors That Affect Roots of Equation Calculator Results

1. Value of 'a': Determines if the parabola opens upwards (a>0) or downwards (a<0) and its width. It cannot be zero. Changing 'a' significantly alters the roots.
2. Value of 'b': Shifts the axis of symmetry and the vertex of the parabola horizontally, thus affecting the roots.
3. Value of 'c': This is the y-intercept of the parabola. Changing 'c' shifts the parabola vertically, directly impacting the values of the roots and whether they are real or complex.
4. The Discriminant (b² – 4ac): This value, derived from a, b, and c, is the most direct indicator of the nature of the roots (real and distinct, real and equal, or complex).
5. Sign of 'a' and 'c': If 'a' and 'c' have opposite signs, the discriminant is always positive (b² – 4ac = b² – (negative) = positive), guaranteeing two real roots.
6. Magnitude of 'b' relative to '4ac': If b² is much larger than |4ac|, the roots are real and far apart. If b² is close to 4ac, the roots are close together or equal. If b² is less than 4ac (and 4ac>0), the roots are complex.

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.

Why can't 'a' be zero in the Roots of Equation Calculator?

If 'a' is zero, the ax² term disappears, and the equation becomes bx + c = 0, which is a linear equation, not quadratic. Our Roots of Equation Calculator is specifically for quadratic equations.

What does the discriminant tell me?

The discriminant (Δ = b² – 4ac) tells you the nature of the roots: Δ > 0 means two distinct real roots, Δ = 0 means one real root (or two equal real roots), and Δ < 0 means two complex conjugate roots.

What are complex roots?

Complex roots occur when the discriminant is negative. They involve the imaginary unit 'i' (where i² = -1) and are expressed in the form p ± qi.

How many roots does a quadratic equation have?

A quadratic equation always has two roots, according to the fundamental theorem of algebra. These roots can be real and distinct, real and equal, or complex conjugates.

Can I use this Roots of Equation Calculator for cubic equations?

No, this calculator is specifically designed for quadratic equations (degree 2). Cubic equations (degree 3) have different solution methods.

What does it mean if the roots are equal?

If the roots are equal, it means the vertex of the parabola y = ax² + bx + c lies exactly on the x-axis. The discriminant is zero.

How accurate is this Roots of Equation Calculator?

The calculator uses standard mathematical formulas and is very accurate, limited only by the precision of the JavaScript number type.