# Quadratic Equation Solver

A **quadratic equation solver** is a free step by step solver for solving the quadratic equation to find the values of the variable. With the help of this solver, we can find the roots of the quadratic equation given by, ax^{2} + bx + c = 0, where the variable x has two roots. The solution is obtained using the quadratic formula;

where a, b and c are the real numbers and a ≠ 0. If a = 0, then the equation becomes linear. We can call it a linear equation. The quadratic equation is of three types namely,

- Standard form
- Factored form
- Vertex form

Generally, there are four different methods to solve the quadratic equation. Those methods are:

- Factoring
- Using square roots
- Completing the squares
- Using quadratic formula

In this, quadratic equation solver page, we will use the quadratic formula to solve the quadratic equation.

## How does the Quadratic Equation Solver Work?

A quadratic equation is nothing but a polynomial of degree 2. The roots of polynomials give the solution of the equation. Here we have to solve an equation in the form of ax^{2} + bx + c = 0.

The quadratic equation solver uses the quadratic formula to find the roots of the given quadratic equation. The procedure to use the quadratic equation solver is as follows:

**Step 1: **Enter the coefficients of the quadratic equation “a”, “b” and “c” in the input fields.

**Step 2: **Now, click the button “Solve the Quadratic Equation” to get the roots.

**Step 3: **Finally, the discriminant and the roots of the given quadratic equation will be displayed in the output fields.

Enter the values of a, b and c in the solver given below to solve any given quadratic equation.

## Steps to Solve Quadratic Equation

The input for the quadratic equation solver is of the form

ax^{2} + bx + c = 0

Where a is not zero, a ≠ 0

If the value of a is zero, then the equation is not a quadratic equation.

The quadratic equation solution is obtained using the quadratic formula:

\(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\)Normally, we get two solutions, because of a plus or minus symbol “±”. You need to do both the addition and subtraction operation.

The part of an equation “ b^{2}-4ac “ is called the “**discriminant**” and it produces the different types of possible solutions. Some of the possible solutions are

- Case 1: When a discriminant part is positive, you get two real solutions
- Case 2: When a discriminant part is zero, it gives only one solution
- Case 3: When a discriminant part is negative, you get complex solutions

Quadratic solver level helps the students of class 10 to clearly know about the different cases involved in the discriminant producing different solutions. Here are some of the quadratic equation examples

### Quadratic Formula Examples

**Case 1 : b**^{2}– 4ac > 0

**Example 1: Consider an example x ^{2} – 3x – 10 = 0**

Given data : a =1, b = -3 and c = -10

b^{2} – 4ac = (-3)^{2}– 4 (1)(-10)

= 9 +40 = 49

b^{2} – 4ac= 49 >0

Therefore, we get **two real solutions**

The general quadratic formula is given as;

\(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\) \(x=\frac{-(-3)\pm \sqrt{(-3)^{2}-4(1)(-10)}}{2(1)}\) \(x=\frac{3\pm \sqrt{9+40}}{2}\) \(x=\frac{3\pm \sqrt{49}}{2}\) \(x=\frac{3\pm 7}{2}\)x= 10/2 , -4/2

x= 5, -2

Therefore, the solutions are 5 and -2

**Case 2 : b**^{2}– 4ac = 0

**Example 2: Consider an example 9x ^{2 }+12x + 4 = 0**

Given data : a =9, b = 12 and c = 4

b^{2} – 4ac = (12)^{2}– 4 (9)(4)

= 144 – 144= 0

b^{2} – 4ac= 0

Therefore, we get only **one distinct solution**

The general quadratic formula is given as

\(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\) \(x=\frac{-(12)\pm \sqrt{(12)^{2}-4(9)(4)}}{2(9)}\) \(x=\frac{-12\pm \sqrt{144-144}}{18}\) \(x=\frac{-12\pm \sqrt{0}}{18}\) \(x=\frac{-12}{18}\)x= -6/9 = -2/3

x= -2/3

Therefore, the solution is -2 / 3

**Case 3 : b**^{2}– 4ac < 0

**Example 3: Consider an example x ^{2} + x + 12= 0**

Given data : a =1, b = 1 and c = 12

b^{2} – 4ac = (1)^{2}– 4 (1)(12)

= 1 – 48 = -47

b^{2} – 4ac= -47 < 0

Therefore, we get **complex solutions**

The general quadratic formula is given as

\(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\) \(x=\frac{-(1)\pm \sqrt{(1)^{2}-4(1)(12)}}{2(1)}\) \(x=\frac{-1\pm \sqrt{1-48}}{2}\) \(x=\frac{-1\pm \sqrt{-47}}{2}\) \(x=\frac{-1+i\sqrt{47}}{2}\) and \(x=\frac{-1-i\sqrt{47}}{2}\)Therefore, the solutions are

\(x=\frac{-1+i\sqrt{47}}{2}\) and \(x=\frac{-1-i\sqrt{47}}{2}\)For more information about quadratic equations and other related topics in mathematics, register with BYJU’S – The Learning App and watch interactive videos.

## Frequently Asked Questions on Quadratic Equation Solver

### What is meant by the quadratic equation?

In Maths, the quadratic equation is defined as an algebraic equation of degree 2, and it should be in the form of ax^{2} + bx + c = 0. Here, a, b, and c are the coefficients of the variable x, and the value of “a” should not be equal to 0. (i.e., a≠ 0). The solutions of the quadratic equation are called the roots of the equation.

### What are the four different methods to solve the quadratic equation?

The different methods to solve the quadratic equation are:

Factoring

Completing the squares

Using the square root method

Quadratic formula

### What is discriminant?

The discriminant D = b^{2} – 4ac reveals the nature of the roots that the equation has. It is determined from the coefficients of the equation.

If D = 0, the roots are equal, real and rational

If D > 0, and also a perfect square, the roots are real, distinct and rational

If D > 0, but not a perfect square, the roots are real, distinct and irrational

### What is the standard form of the quadratic equation?

The standard form to represent the quadratic equation is

Ax^{2} + Bx + C = 0

Here A, B and C are the known values, and A should not be equal to 0.

X is a variable.

### Mention the applications of quadratic equations.

The quadratic equations are used in everyday life activities such as finding the profit of the product, calculating the area of the room, athletics, finding the speed of the object, and so on.