November 24, 2022

Quadratic Equation Formula, Examples

If this is your first try to work on quadratic equations, we are excited regarding your journey in mathematics! This is actually where the most interesting things begins!

The data can look overwhelming at first. But, provide yourself some grace and space so there’s no hurry or stress when working through these questions. To be efficient at quadratic equations like an expert, you will need a good sense of humor, patience, and good understanding.

Now, let’s begin learning!

What Is the Quadratic Equation?

At its center, a quadratic equation is a mathematical equation that portrays distinct situations in which the rate of change is quadratic or relative to the square of some variable.

Though it seems like an abstract theory, it is just an algebraic equation stated like a linear equation. It generally has two results and uses intricate roots to figure out them, one positive root and one negative, through the quadratic formula. Solving both the roots the answer to which will be zero.

Definition of a Quadratic Equation

Foremost, remember that a quadratic expression is a polynomial equation that consist of a quadratic function. It is a second-degree equation, and its usual form is:

ax2 + bx + c

Where “a,” “b,” and “c” are variables. We can utilize this equation to solve for x if we replace these terms into the quadratic formula! (We’ll subsequently check it.)

Ever quadratic equations can be scripted like this, which results in solving them easy, comparatively speaking.

Example of a quadratic equation

Let’s contrast the ensuing equation to the previous equation:

x2 + 5x + 6 = 0

As we can see, there are 2 variables and an independent term, and one of the variables is squared. Thus, compared to the quadratic formula, we can assuredly say this is a quadratic equation.

Usually, you can see these kinds of formulas when measuring a parabola, that is a U-shaped curve that can be graphed on an XY axis with the details that a quadratic equation offers us.

Now that we know what quadratic equations are and what they look like, let’s move forward to solving them.

How to Solve a Quadratic Equation Utilizing the Quadratic Formula

Although quadratic equations may seem very complicated when starting, they can be cut down into several easy steps employing a simple formula. The formula for figuring out quadratic equations consists of setting the equal terms and using rudimental algebraic functions like multiplication and division to obtain two solutions.

Once all functions have been performed, we can figure out the units of the variable. The answer take us another step nearer to find solutions to our actual problem.

Steps to Working on a Quadratic Equation Utilizing the Quadratic Formula

Let’s promptly plug in the original quadratic equation again so we don’t overlook what it seems like

ax2 + bx + c=0

Prior to figuring out anything, keep in mind to separate the variables on one side of the equation. Here are the three steps to figuring out a quadratic equation.

Step 1: Write the equation in standard mode.

If there are variables on both sides of the equation, total all alike terms on one side, so the left-hand side of the equation is equivalent to zero, just like the conventional model of a quadratic equation.

Step 2: Factor the equation if feasible

The standard equation you will wind up with must be factored, generally using the perfect square process. If it isn’t possible, put the variables in the quadratic formula, which will be your closest friend for working out quadratic equations. The quadratic formula looks similar to this:


Every terms correspond to the identical terms in a conventional form of a quadratic equation. You’ll be employing this a lot, so it is wise to remember it.

Step 3: Apply the zero product rule and figure out the linear equation to discard possibilities.

Now once you possess 2 terms equivalent to zero, figure out them to attain two answers for x. We possess two results due to the fact that the answer for a square root can either be negative or positive.

Example 1

2x2 + 4x - x2 = 5

Now, let’s piece down this equation. Primarily, streamline and place it in the conventional form.

x2 + 4x - 5 = 0

Now, let's determine the terms. If we contrast these to a standard quadratic equation, we will get the coefficients of x as follows:




To work out quadratic equations, let's put this into the quadratic formula and find the solution “+/-” to include each square root.



We figure out the second-degree equation to get:



After this, let’s clarify the square root to attain two linear equations and work out:

x=-4+62 x=-4-62

x = 1 x = -5

Next, you have your result! You can review your workings by using these terms with the first equation.

12 + (4*1) - 5 = 0

1 + 4 - 5 = 0


-52 + (4*-5) - 5 = 0

25 - 20 - 5 = 0

That's it! You've figured out your first quadratic equation using the quadratic formula! Congrats!

Example 2

Let's try one more example.

3x2 + 13x = 10

First, place it in the standard form so it results in 0.

3x2 + 13x - 10 = 0

To work on this, we will substitute in the figures like this:

a = 3

b = 13

c = -10

Work out x employing the quadratic formula!



Let’s simplify this as far as feasible by working it out just like we performed in the prior example. Solve all simple equations step by step.



You can solve for x by considering the positive and negative square roots.

x=-13+176 x=-13-176

x=46 x=-306

x=23 x=-5

Now, you have your solution! You can review your workings through substitution.

3*(2/3)2 + (13*2/3) - 10 = 0

4/3 + 26/3 - 10 = 0

30/3 - 10 = 0

10 - 10 = 0


3*-52 + (13*-5) - 10 = 0

75 - 65 - 10 =0

And this is it! You will work out quadratic equations like a pro with some patience and practice!

Given this overview of quadratic equations and their fundamental formula, students can now take on this complex topic with assurance. By opening with this simple explanation, learners secure a strong grasp prior undertaking more complicated theories down in their academics.

Grade Potential Can Assist You with the Quadratic Equation

If you are fighting to understand these concepts, you might need a mathematics teacher to assist you. It is best to ask for guidance before you get behind.

With Grade Potential, you can learn all the tips and tricks to ace your next math examination. Turn into a confident quadratic equation problem solver so you are ready for the ensuing big theories in your math studies.