Quadratic Equation Solver

A quadratic equation is a very common part of mathematics. It is used in schools, colleges, and real life. It looks simple but has many uses. Let us learn it step by step.

What is a Quadratic Equation?

A quadratic equation is an equation of degree two. It means the highest power of the variable is two.

Quadratic equations are expressed in the form ax^2 + bx + c = 0.

a, b, c are numbers.
a cannot be zero.
x is the unknown value.

Example: 2x^2 + 3x - 5 = 0

Parts of a Quadratic Equation

Every quadratic equation has three main parts:
Coefficient of x^2 → This is a.
Coefficient of x → This is b.
Constant term → This is c.

In 2x^2 + 3x - 5 = 0:
a = 2
b = 3
c = -5

Roots of a Quadratic Equation

The solution of a quadratic equation is called a root. A quadratic equation always has two roots.

The roots can be:
Two different real numbers
Two equal real numbers
Two complex numbers

Methods to Solve Quadratic Equations

Factoring Method

We write the quadratic as a product of two simple brackets.
Example: x^2 - 7x + 12 = 0
(x-3)(x-4) = 0
So, x = 3 or x = 4.

Quadratic Formula

We use a direct formula:
x = (-b ± √(b^2 - 4ac)) / 2a

The part under the root sign is called the discriminant.
If discriminant is greater than 0 → two real roots.
If discriminant is equal to 0 → one real root (repeated).
If discriminant is less than 0 → two complex roots.

Completing the Square

We make the quadratic into a perfect square.
Example: x^2 + 4x + 1 = 0
x^2 + 4x = -1
x^2 + 4x + 4 = 3
(x+2)^2 = 3
So, x+2 = ±√3
Therefore, x = -2+√3 or x = -2-√3

Graph of a Quadratic Equation

A quadratic equation can be shown on a graph. The graph is always a parabola.
If a > 0, the parabola opens upward.
If a < 0>

The vertex of a parabola represents its maximum or minimum point. A parabola reaches its highest or lowest value at the vertex. The vertex is the point where the parabola turns, either at the top or the bottom. In a parabola, the vertex shows the peak or the bottommost position. The vertex is the extreme point of the parabola, either maximum or minimum. At the vertex, the parabola has its greatest or least value. The vertex marks the turning point of the parabola, where it is either highest or lowest. The maximum or minimum point of a parabola is called the vertex.

Applications of Quadratic Equations

A class of quadratic equations appear from various practical problems.
Physics → Movement of objects, projectile trajectory.
Engineering → Bridgework, arches, and machines.
Enterprise → Profit and loss calculation.
Geometry → Area and perimeter problems.
Life in the Fast Lane → Speed, time, and distance problems.

Important Terms in Quadratic Equations

Discriminant (D) = b^2 - 4ac.
Vertex = The point on the parabola at which it will be at the highest or lowest.
Axis of symmetry = Vertical line through the vertex.
Parabola = The graph’s curve.

Conclusion

A quadratic equation is an important topic in algebra. It looks simple but it has many uses. It can be solved in different ways. It also has many applications in science, business, and daily life. Learning quadratic equations helps to build strong math skills.