Quadratic Equation Solver Online

We present to you an online calculator and application for solving quadratic equations: the Online Quadratic Equation Solver. In this article, we’ll discuss the formula and method for solving the equation, along with many solved examples and exercises. The general form of the mathematical equation is as follows:

• The values a, b, and c are called the quadratic equation’s coefficients.
• Where C is a numerical constant.
• b is the coefficient of the first-degree variable x.
• a is the coefficient of the second-degree variable x².
• The fundamental condition for the equation is that a is not equal to zero (a≠0).
• The goal of solving the equation is to find the correct possible values of x for which the equation is true.

You can also check out Solving a Quadratic Equation by Completing the Square, as well as Solving a Quadratic Equation by Factoring.

Online Quadratic Equation Solver Program

Below is a program to solve a quadratic equation. Enter the values of a, b, and c and press the Solve Equation button to find the set of solutions for the quadratic equation, in addition to graphing the corresponding function (the graph is a trial version and does not plot the imaginary function where the value of Delta is less than zero).

How to Solve a Quadratic Equation (using the general formula)

The method for solving a quadratic equation is summarized in the following steps:

Steps to solve a quadratic equation:

1. Find Delta First, we find Delta, which is determined by the equation:
   
2. Determine the nature of the roots based on the value of the Discriminant We distinguish 3 cases for the values of x based on the value of Delta:
   1. Delta is greater than zero △>0 : The equation has two real roots.
   2. Delta is less than zero △<0 : The equation has two complex roots.
   3. Delta is equal to zero △=0 : The equation has one single root.
   
3. Case 1: Delta is greater than zero △>0 The value of the two real roots of the equation is calculated according to the formula:
   
   The presence of the ± sign means you must perform two operations, addition for the first root and subtraction for the other.
4. Case 2: Delta is less than zero △<0 The equation has two imaginary roots, each consisting of a real part and an imaginary part. The two roots are calculated according to the formula:
   
5. Case 3: Delta is equal to zero (△=0) The equation has one unique solution, a double root, whose value is determined by the formula:
   

Quadratic Equation Exercises

We offer a variety of exercises on solving quadratic equations. If you want more, you can check out our article Quadratic Equation Exercises, where we have dedicated many unique exercises for you.

Exercise One

Let’s have the following equation:

x²+2x-3 = 0

Find all values of x

Solution: We note that a=1, b=2, c=-3. The general formula for the set of solutions for a quadratic equation is given by the two relations:

x1 = (-b+√Delta)/2a
x2 = (-b-√Delta)/2a

While the discriminant Delta relation is given by the following equation:

Delta = b²-4ac

We substitute the values of a, b, and c from the equation into the discriminant equation:

deta = 2²-4×1×(-3)
= 16

Delta (the discriminant) is greater than zero, so the equation has two solutions or roots.

x1 =(-2+√16)/2×1 = 1
x2 =(-2-√16)/2×1 = -3

Exercise Two

Find the solutions to the following quadratic equation:

8x²-64 = 0

Solution: We find that a=8, b=0, c=-64. Let’s calculate the discriminant Delta.

delta = b²-4ac
delta =(0)²-4×8×(-64)

Delta is greater than zero, so the equation also has two roots.

x1 = +√(4×8×64)/2×8
x1 = +√(4×4×2×64)/2×8
x1 = +√(4×4×2×64)/2×8
x1 = +32√2/16

Thus, the solutions to the equation are:

x1 = +2√2
x2 = -2√2

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Exercise Three

Let’s have the following equation:

6 x²+4x = 2x²-1

Find all values of x

Solution: First, we rearrange the equation:

6x²-2x²+4x+1 = 0
4x²+4x+1 = 0

Now let’s find the value of the discriminant Delta:

delta = 4²-4×4×1
16-16 = 0

Delta is equal to zero, and the equation has a single double root.

x = -b/2a
x = -4/2×4
= -0.5

You may also be interested in: Solving a Cubic Equation Online (Cubic Equation Solver)

Frequently Asked Questions about Quadratic Equations