November 24, 2022

Quadratic Equation Formula, Examples

If you’re starting to work on quadratic equations, we are thrilled about your adventure in mathematics! This is really where the amusing part starts!

The information can look too much at first. Despite that, provide yourself some grace and space so there’s no pressure or strain while solving these problems. To be efficient at quadratic equations like a pro, you will require a good sense of humor, patience, and good understanding.

Now, let’s start learning!

What Is the Quadratic Equation?

At its center, a quadratic equation is a arithmetic formula that describes distinct situations in which the rate of change is quadratic or relative to the square of few variable.

Although it seems like an abstract idea, it is just an algebraic equation expressed like a linear equation. It usually has two answers and utilizes complex roots to solve them, one positive root and one negative, through the quadratic formula. Solving both the roots should equal zero.

Definition of a Quadratic Equation

Primarily, bear 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 utilize this equation to work out x if we replace these variables into the quadratic formula! (We’ll get to that later.)

Any quadratic equations can be written like this, that results in figuring them out easy, relatively speaking.

Example of a quadratic equation

Let’s contrast the following equation to the subsequent equation:

x2 + 5x + 6 = 0

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

Commonly, you can observe these kinds of equations when scaling a parabola, that is a U-shaped curve that can be plotted on an XY axis with the information that a quadratic equation offers us.

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

How to Solve a Quadratic Equation Using the Quadratic Formula

Even though quadratic equations may look greatly complicated initially, they can be broken down into few easy steps utilizing a straightforward formula. The formula for figuring out quadratic equations consists of creating the equal terms and using rudimental algebraic functions like multiplication and division to achieve two answers.

Once all functions have been performed, we can work out the values of the variable. The results take us another step nearer to discover answer to our first problem.

Steps to Working on a Quadratic Equation Employing the Quadratic Formula

Let’s quickly plug in the original quadratic equation again so we don’t forget 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 work on a quadratic equation.

Step 1: Write the equation in standard mode.

If there are variables on either side of the equation, add all alike terms on one side, so the left-hand side of the equation equals zero, just like the conventional mode of a quadratic equation.

Step 2: Factor the equation if workable

The standard equation you will conclude with should be factored, ordinarily utilizing the perfect square method. If it isn’t workable, plug the variables in the quadratic formula, that will be your best friend for figuring out quadratic equations. The quadratic formula looks something like this:


All the terms responds to the equivalent terms in a standard form of a quadratic equation. You’ll be employing this a lot, so it pays to remember it.

Step 3: Implement the zero product rule and work out the linear equation to eliminate possibilities.

Now once you possess two terms equivalent to zero, work on them to obtain two solutions for x. We possess two results because 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. First, streamline and place it in the conventional form.

x2 + 4x - 5 = 0

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




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



We work on the second-degree equation to obtain:



Next, let’s simplify the square root to obtain two linear equations and solve:

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

x = 1 x = -5

Next, 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 using the quadratic formula! Congrats!

Example 2

Let's try one more example.

3x2 + 13x = 10

Let’s begin, place it in the standard form so it results in zero.

3x2 + 13x - 10 = 0

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

a = 3

b = 13

c = -10

Solve for x utilizing the quadratic formula!



Let’s clarify this as much as workable by figuring it out exactly like we did in the prior example. Work out all easy equations step by step.



You can solve for x by considering the negative and positive 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 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 professional with some practice and patience!

Granted this synopsis of quadratic equations and their rudimental formula, children can now take on this complex topic with assurance. By beginning with this easy explanation, learners gain a strong understanding before moving on to more complicated theories ahead in their studies.

Grade Potential Can Guide You with the Quadratic Equation

If you are fighting to understand these theories, you may need a mathematics instructor to help you. It is best to ask for assistance before you fall behind.

With Grade Potential, you can learn all the handy tricks to ace your next math test. Become a confident quadratic equation solver so you are prepared for the following big theories in your mathematics studies.