Introduction

Algebraically solving quadratic equations is fundamental in mathematics and is a basis for tackling different types of problems both in academics and real life. A typical quadratic equation takes this form:

ax² + bx + c = 0

Where;

• a, b, c are constants.
• x represents the variable.

Even though there are various ways through which a quadratic equation can be solved such as factoring, completing the square or using the quadratic formula; these calculations may sometimes become tiresome especially when it comes to complex equations. Luckily enough, the task is simplified by Quadratic Equation Solver at Calculator3.com that provides accurate answers promptly for both real and complex roots.

This article will bring out how the Quadratic Equation Solver operates, alternative methods to solve a quadratic equation as well as its significance in terms of making you work better on your equations.

Quadratic Equation: Definition

A quadratic equation is a second-degree polynomial equation that can be written as follows:

ax² + bx + c = 0

In the above equation:

• a, b, and c are constants, and a ≠ 0.

The power of x in this case is raised to two, making it an example of a parabola.

Quadratic equations can have:

• If the discriminant (b² − 4ac) is positive, then they will have two real roots.
• When the discriminant equals zero (a repeated root), they will only have one real solution.
• If the discriminant is negative, then there will be two complex solutions for such equations.

Techniques of Solving Quadratic Equations

Different quadratic equations have different approaches used to solve them. For these approaches, Calculator3.com’s Quadratic Equation Solver can be used:

1. Factoring

Factoring is a method of making the quadratic equation as a product of two binomials, e.g., x² – 5x + 6 = 0:

(x – 2)(x – 3) = 0.

This gives two solutions namely: x = 2 and x = 3.

Factoring is easy when you have simple or easily factorable quadratic equations. However, there are some quadratic equations that cannot be factored, such as those whose roots are not rational numbers or complex numbers.

2. Completion of the Square

Completing the square is a way of converting a quadratic equation into a perfect square trinomial. For example, if we take this equation:

x² + 6x = 7,

it can be rewritten by adding constants to both sides to yield a perfect square trinomial as shown below:

(x + 3)² = 16.

After solving for x, we get:

(−3±4), or x = 1 and x = −7.

3. The Quadratic Formula

The most general method for solving quadratic equations, whether easily factored or not, is represented by the quadratic formula. It is given by:

x = −b ± √(b² − 4ac) / 2a

Where:

• a, b, and c are coefficients from the quadratic equation ax² + bx + c = 0.

The discriminant (b² − 4ac) tells us about the roots’ nature as follows:

• Two real solutions exist when Δ > 0.
• A discriminant of zero implies only one real solution (repeated root).
• A negative discriminant gives two complex roots.

Using Calculator3.com’s Quadratic Equation Solver

The Quadratic Equation Solver from Calculator3.com simplifies the quadratic equation solving process, which saves time and minimizes possible mistakes. Below is how to use it:

Input Coefficients

Type in the number values of a, b, and c from your quadratic equation. For example, in 2x² − 4x − 6 = 0, you should type:

• a = 2
• b = −4
• c = −6

Choose Method

The calculator defaults to using the quadratic formula to solve the equation, but you can also select other methods such as factoring or completing the square, depending on your preferences or requirements.

Click “Solve”

Once you have entered coefficients above, click on the “Solve” button. It will calculate solutions for you.

Results

The results include whether or not the roots are complex or real, their discriminant value, and some detailed steps where applicable.

Example Walkthrough: Solving a Quadratic Equation

As an example, we will solve the quadratic equation by using the Quadratic Equation Solver. Consider the equation:

x² − 5x + 6 = 0

Input the coefficients:

• a = 1
• b = −5
• c = 6

Click “Solve”: The calculator will use the quadratic formula to find the solutions.

Solution: The result will show:

The discriminant: Δ = (−5)² − 4(1)(6) = 25 − 24 = 1

Thus, the solutions are x = 3 and x = 2.

Applications of Quadratic Equations

• Physics: Projectile motion, falling objects, and other forms of motion can be modeled using quadratic equations.
• Finance: Quadratic equations can be used for modeling optimization problems like maximization of profit or minimization of costs.
• Engineering: They help in designing structures and analyzing forces and stresses.
• Biology: Quadratic equations are useful models for population growth and other biological processes.