November 24, 2022

Quadratic Equation Formula, Examples

If you’re starting to solve quadratic equations, we are excited regarding your venture in math! This is actually where the fun begins!

The information can appear too much at start. But, provide yourself some grace and room so there’s no hurry or stress when figuring out these problems. To be competent at quadratic equations like a pro, you will require patience, understanding, and a sense of humor.

Now, let’s start learning!

What Is the Quadratic Equation?

At its core, a quadratic equation is a arithmetic equation that states distinct scenarios in which the rate of deviation is quadratic or proportional to the square of few variable.

Although it may look like an abstract concept, it is just an algebraic equation stated like a linear equation. It usually has two answers and utilizes complex roots to solve them, one positive root and one negative, using the quadratic formula. Working out both the roots will be equal to zero.

Definition of a Quadratic Equation

Foremost, keep in mind that a quadratic expression is a polynomial equation that consist of a quadratic function. It is a second-degree equation, and its standard form is:

ax2 + bx + c

Where “a,” “b,” and “c” are variables. We can employ this equation to work out x if we plug these numbers into the quadratic equation! (We’ll get to that later.)

Ever quadratic equations can be scripted like this, that makes working them out simply, relatively speaking.

Example of a quadratic equation

Let’s compare the following equation to the subsequent formula:

x2 + 5x + 6 = 0

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

Commonly, you can observe these kinds of formulas when scaling a parabola, that is a U-shaped curve that can be graphed on an XY axis with the details that a quadratic equation gives us.

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

How to Figure out a Quadratic Equation Using the Quadratic Formula

Although quadratic equations might look greatly intricate initially, they can be cut down into multiple simple steps employing an easy formula. The formula for solving quadratic equations consists of creating the equal terms and utilizing fundamental algebraic operations like multiplication and division to achieve 2 answers.

After all operations have been executed, we can figure out the units of the variable. The answer take us another step closer to discover answer to our original question.

Steps to Working on a Quadratic Equation Using the Quadratic Formula

Let’s promptly put in the original quadratic equation again so we don’t forget what it looks like

ax2 + bx + c=0

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

Step 1: Note the equation in conventional mode.

If there are variables on either side of the equation, total all similar terms on one side, so the left-hand side of the equation totals to zero, just like the conventional model of a quadratic equation.

Step 2: Factor the equation if possible

The standard equation you will conclude with should be factored, generally utilizing the perfect square method. If it isn’t workable, replace the terms in the quadratic formula, that will be your closest friend for solving quadratic equations. The quadratic formula seems like this:


All the terms correspond to the equivalent terms in a standard form of a quadratic equation. You’ll be employing this a great deal, so it is wise to remember it.

Step 3: Implement the zero product rule and solve the linear equation to discard possibilities.

Now that you possess 2 terms resulting in zero, solve them to get 2 results for x. We get two answers because the answer for a square root can be both negative or positive.

Example 1

2x2 + 4x - x2 = 5

At the moment, let’s break down this equation. Primarily, simplify and put it in the conventional form.

x2 + 4x - 5 = 0

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




To solve quadratic equations, let's replace this into the quadratic formula and solve for “+/-” to involve each square root.



We figure out the second-degree equation to get:



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

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

x = 1 x = -5

After that, you have your solution! You can check your work by using these terms with the initial 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 utilizing the quadratic formula! Congratulations!

Example 2

Let's work on another example.

3x2 + 13x = 10

First, put it in the standard form so it equals zero.

3x2 + 13x - 10 = 0

To figure out this, we will substitute in the values like this:

a = 3

b = 13

c = -10

figure out x using the quadratic formula!



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



You can work out 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 revise your work using 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 that's it! You will figure out quadratic equations like a pro with some patience and practice!

Granted this summary of quadratic equations and their basic formula, students can now take on this difficult topic with faith. By starting with this easy definitions, learners secure a strong grasp ahead of taking on further intricate concepts later in their academics.

Grade Potential Can Guide You with the Quadratic Equation

If you are fighting to understand these theories, you might require a math tutor to help you. It is better to ask for help before you get behind.

With Grade Potential, you can learn all the tips and tricks to ace your subsequent math test. Become a confident quadratic equation solver so you are ready for the ensuing big ideas in your math studies.