November 24, 2022

Quadratic Equation Formula, Examples

If you’re starting to solve quadratic equations, we are enthusiastic regarding your venture in mathematics! This is really where the fun starts!

The data can appear too much at first. But, offer yourself some grace and space so there’s no pressure or stress while solving these problems. To be efficient at quadratic equations like a pro, you will need patience, understanding, and a sense of humor.

Now, let’s start learning!

What Is the Quadratic Equation?

At its heart, a quadratic equation is a math formula that describes various scenarios in which the rate of change is quadratic or relative to the square of some variable.

Although it might appear like an abstract concept, it is simply an algebraic equation described like a linear equation. It usually has two results and utilizes complicated roots to solve them, one positive root and one negative, employing the quadratic formula. Solving both the roots will be equal to zero.

Definition of a Quadratic Equation

First, keep in mind that a quadratic expression is a polynomial equation that includes a quadratic function. It is a second-degree equation, and its conventional form is:

ax2 + bx + c

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

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

Example of a quadratic equation

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

x2 + 5x + 6 = 0

As we can observe, there are two variables and an independent term, and one of the variables is squared. Consequently, compared to the quadratic formula, we can surely state this is a quadratic equation.

Usually, you can find these types of equations 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 provides us.

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

How to Work on a Quadratic Equation Employing the Quadratic Formula

While quadratic equations may appear greatly complicated when starting, they can be divided into several easy steps using a simple formula. The formula for working out quadratic equations consists of setting the equal terms and applying basic algebraic functions like multiplication and division to achieve 2 answers.

After all functions have been carried out, we can work out the units of the variable. The solution take us another step closer to discover answer to our original problem.

Steps to Figuring out a Quadratic Equation Employing the Quadratic Formula

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

ax2 + bx + c=0

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

Step 1: Write the equation in conventional 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 totals to zero, just like the standard mode of a quadratic equation.

Step 2: Factor the equation if workable

The standard equation you will wind up with should be factored, ordinarily through the perfect square process. If it isn’t feasible, put the terms in the quadratic formula, that will be your best buddy for working out quadratic equations. The quadratic formula looks like this:


All the terms responds to the same terms in a conventional form of a quadratic equation. You’ll be employing this a lot, so it is wise to memorize it.

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

Now once you have two terms equivalent to zero, solve them to achieve two results for x. We possess 2 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, clarify and place it in the conventional form.

x2 + 4x - 5 = 0

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




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



We solve the second-degree equation to achieve:



Now, let’s clarify the square root to obtain two linear equations and solve:

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

x = 1 x = -5

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

12 + (4*1) - 5 = 0

1 + 4 - 5 = 0


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

25 - 20 - 5 = 0

This is it! You've solved your first quadratic equation using the quadratic formula! Congratulations!

Example 2

Let's work on one more example.

3x2 + 13x = 10

Let’s begin, put it in the standard form so it is equivalent 0.

3x2 + 13x - 10 = 0

To solve this, we will substitute in the values like this:

a = 3

b = 13

c = -10

Solve for x utilizing the quadratic formula!



Let’s simplify this as much as possible by figuring it out just like we did in the prior example. Figure 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 result! You can revise your workings 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 solve quadratic equations like a pro with a bit of practice and patience!

With this summary of quadratic equations and their fundamental formula, children can now take on this complex topic with assurance. By opening with this simple definitions, children secure a firm grasp prior taking on further complicated theories later in their studies.

Grade Potential Can Assist You with the Quadratic Equation

If you are struggling to get a grasp these theories, you may need a mathematics tutor to guide you. It is best to ask for help before you fall behind.

With Grade Potential, you can learn all the tips and tricks to ace your next math test. Become a confident quadratic equation problem solver so you are prepared for the following big ideas in your mathematical studies.