November 24, 2022

Quadratic Equation Formula, Examples

If you’re starting to figure out quadratic equations, we are thrilled regarding your journey in math! This is indeed where the fun starts!

The details can look overwhelming at first. However, offer yourself some grace and room so there’s no hurry or stress when working through these questions. To be competent at quadratic equations like an expert, you will require understanding, patience, 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 describes distinct situations in which the rate of deviation is quadratic or relative to the square of some variable.

Although it might appear like an abstract idea, it is simply an algebraic equation described like a linear equation. It generally has two answers and uses intricate roots to work out them, one positive root and one negative, using the quadratic equation. Working out both the roots the answer to which will be zero.

Definition of a Quadratic Equation

First, 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 use this formula to figure out x if we put these variables into the quadratic equation! (We’ll look at it next.)

All quadratic equations can be scripted like this, which makes figuring them out straightforward, comparatively speaking.

Example of a quadratic equation

Let’s contrast 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 surely tell this is a quadratic equation.

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

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

How to Solve a Quadratic Equation Employing the Quadratic Formula

Although quadratic equations might appear very complex initially, they can be divided into multiple simple steps employing an easy formula. The formula for solving quadratic equations involves creating the equal terms and using rudimental algebraic functions like multiplication and division to get 2 results.

Once all functions have been performed, we can work out the values of the variable. The results take us single step closer to work out the answer to our actual problem.

Steps to Figuring out a Quadratic Equation Using the Quadratic Formula

Let’s quickly place in the general quadratic equation once more so we don’t forget what it seems like

ax2 + bx + c=0

Before solving anything, remember to isolate the variables on one side of the equation. Here are the 3 steps to solve a quadratic equation.

Step 1: Note the equation in conventional mode.

If there are variables on both sides of the equation, add all similar terms on one side, so the left-hand side of the equation equals zero, just like the standard mode of a quadratic equation.

Step 2: Factor the equation if possible

The standard equation you will end up with must be factored, generally using the perfect square process. If it isn’t possible, put the variables in the quadratic formula, that will be your best friend for working out quadratic equations. The quadratic formula seems like this:


All the terms responds to the same terms in a conventional 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 remove possibilities.

Now that you possess 2 terms resulting in zero, solve them to obtain 2 results for x. We have two results due to the fact that the solution for a square root can be both positive or negative.

Example 1

2x2 + 4x - x2 = 5

At the moment, let’s break down this equation. First, streamline and put it in the standard form.

x2 + 4x - 5 = 0

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




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



We work on the second-degree equation to obtain:



Next, let’s simplify the square root to achieve two linear equations and work out:

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

x = 1 x = -5

After that, you have your answers! You can review your workings by checking 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! Congrats!

Example 2

Let's check out another example.

3x2 + 13x = 10

Initially, place it in the standard form so it results in zero.

3x2 + 13x - 10 = 0

To work on this, we will put in the values like this:

a = 3

b = 13

c = -10

figure out x utilizing the quadratic formula!



Let’s simplify this as far as possible by working it out exactly like we executed in the prior example. Work out all simple equations step by step.



You can figure 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 check your work 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 figure out quadratic equations like nobody’s business with some practice and patience!

Granted this overview of quadratic equations and their basic formula, children can now tackle this difficult topic with confidence. By opening with this simple explanation, children secure a firm understanding before taking on further complicated ideas ahead in their studies.

Grade Potential Can Guide You with the Quadratic Equation

If you are fighting to get a grasp these concepts, you may require a mathematics tutor to guide you. It is better to ask for guidance before you trail behind.

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