Find Roots Of Quadratic Equation Using Quadratic Formula

A Quadratic Equation is a trinomial (Polynomial with the third degree). To solve such equations, there are a few methods. For example, Completing the square, factoring. But the most popular method used to solve such equations is the method of calculating the discriminant of the equation - the so-called quadratic formula. This formula, as we will see later, gives us an idea about whether this equation has real solutions or does not. And the number of solutions if it has solutions.

The standard form of the quadratic equation

In a previous lesson, we saw that linear equations are written in the form of x, where x and x are two known numbers and x is an unknown number. Solving this equation requires finding the value of the unknown x that makes both sides of the equation equal.

For Quadratic Equations, the matter is slightly different from linear equations, since the unknown, in this case, is raised to the power of 2, and therefore the method of solving will differ from linear equations.

Here's the definition of a quadratic equations.

Note:

Let , and represent real numbers with . The standard form of the quadratic equations is:

examples of quadratic equations with location of coefficients

The Discriminant of a Quadratic Equation

At the beginning of this lesson, we said that the discriminant of the equation determines a very important thing, which is the number of solutions to the equation. So how do we calculate this important mathematical expression? The formula is very simple, you just have to know well the positions of the coefficients , and in the equation.

Here is the rule we use for calculate the discriminant.

Note:

We consider the following quadratic equation.

The discriminant of this equation is:

The Roots of a Quadratic Equation Using the Quadratic Formula

The discriminant of the equation can therefore be a positive, negative or equal to 0. The sign of this discriminant is what will determine the number of solutions to the equation which are three different cases detailed in the following property.

Note:

  • If . The equation has two different real roots:

In short, we write:

  • If . The equation has one real root:
  • If . The equation has no real roots:

Examples of Using the Quadratic Formula to Solve Quadratic Equations

In these examples we will try to take some examples from the three cases to solve a quadratic equation.

Steps to solve the quadratic equation:

  • Locate the coefficients , and (Check the table above).

  • Calculate .

  • Check the sign of .

  • Derive the solutions to the equation from the above property.

Example1:

Solve the equation:

Solution:

  • The coefficients:
  • Calculate :

.

  • The sign of :

So, there two real roots:

Hence, the two roots are:

Example2:

Solve for x the equation:

Solution:

The equation is not in the standard form, because the left side is different from 0. So, we must make this equation in standard form by subtracting 2 from both sides.

is equivalent to:

is equivalent to:

  • The coefficients:
  • Calculate :

.

  • The sign of :

So, there two real roots:

Hence, the two roots are:

Example3:

Solve for x the equation:

Solution:

  • The coefficients:
  • Calculate :

.

  • The sign of :

So, there one real root:

Hence, the one solution is:

Example4:

Solve for x the equation:

Solution:

  • The coefficients:
  • Calculate :

.

  • The sign of :

So, there are no real solutions for this quadratic equation.