Quadratic Equation Solver

Lets first understand, what are quadratic equations and how we can solve the quadratic equations?

A quadratic equation is any equation that can be rearranged in standard form as

$$ax^2+bx+c=0$$

where x is a variable and a(\(\ne 0\)), b and c are coefficients of the quadratic equation.

The values of x that satisfy the quadratic equation are called solutions of the quadratic equation, and roots or zeros of the quadratic equation. A quadratic equation has at most two solutions. If there is no real solution, there are two complex solutions. If there is only one solution, it is a double root. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation

$$ax^2+bx+c=a(x-r_1)(x-r_2)=0$$

where r₁ and r₂ are the solutions for x of the quadratic equation. Completing the square on a quadratic equation in standard form results in the quadratic formula, which expresses the solutions in terms of a, b, and c.

Quadratic Formula

Let the quadratic equation be \(ax^2+bx+c=0\).

We can find the solutions of this quadratic using the formula

$$x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Discrimant of the quadratic equation

In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation and is often represented using an upper case D.

$$D=b^2-4ac$$

A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:

1. If the discriminant is positive, then there are two distinct roots of the quadratic equation

$$r_1=\dfrac{-b+\sqrt{D}}{2a}\hspace{0.2cm}and\hspace{0.2cm}r_2=\dfrac{-b-\sqrt{D}}{2a}$$

2. If the discriminant is zero, then there is exactly one real root of the quadratic equation

$$r_1=r_2=\dfrac{-b}{2a}$$

3. If the discriminant is negative, then there are no real roots of the quadratic equation. Rather, there are two distinct complex roots of the quadratic equation

$$r_1=\dfrac{-b}{2a}+i\dfrac{\sqrt{-D}}{2a}\hspace{0.2cm}and\hspace{0.2cm}r_2=\dfrac{-b}{2a}-i\dfrac{\sqrt{-D}}{2a}$$

which are complex conjugates of each other and here \(i=\sqrt{-1}\).

For solving quadratic equations using above quadratic equation solver, you have to just write the values in the given input boxes and press the calculate button, you will get the result.