November 24, 2022

Quadratic Equation Formula, Examples

If you’re starting to solve quadratic equations, we are thrilled regarding your venture in mathematics! This is actually where the most interesting things begins!

The data can appear enormous at first. But, provide yourself some grace and space so there’s no pressure or strain when figuring out these problems. To master quadratic equations like an expert, you will require a good sense of humor, patience, and good understanding.

Now, let’s start learning!

What Is the Quadratic Equation?

At its core, a quadratic equation is a math equation that describes various situations in which the rate of deviation is quadratic or relative to the square of some variable.

Although it may look like an abstract theory, it is just an algebraic equation described like a linear equation. It generally has two results and utilizes complex roots to figure out them, one positive root and one negative, through the quadratic equation. Working out both the roots will be equal to zero.

Definition of a Quadratic Equation

Primarily, remember 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 utilize this equation to figure out x if we plug these variables into the quadratic formula! (We’ll go through it later.)

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

Example of a quadratic equation

Let’s compare the following 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. Therefore, compared to the quadratic equation, we can assuredly tell this is a quadratic equation.

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

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

How to Work on a Quadratic Equation Employing the Quadratic Formula

Although quadratic equations may seem greatly complicated when starting, they can be divided into multiple easy steps employing a simple formula. The formula for figuring out quadratic equations involves setting the equal terms and applying fundamental algebraic functions like multiplication and division to get 2 solutions.

Once all operations have been performed, we can figure out the numbers of the variable. The answer take us one step nearer to work out the solutions to our actual problem.

Steps to Figuring out a Quadratic Equation Utilizing the Quadratic Formula

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

ax2 + bx + c=0

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

Step 1: Write the equation in conventional mode.

If there are terms on both sides of the equation, sum all equivalent terms on one side, so the left-hand side of the equation totals to zero, just like the standard model of a quadratic equation.

Step 2: Factor the equation if possible

The standard equation you will conclude with should be factored, usually utilizing the perfect square process. If it isn’t possible, replace the terms in the quadratic formula, which will be your closest friend for working out quadratic equations. The quadratic formula looks something like this:


Every terms responds to the equivalent terms in a conventional form of a quadratic equation. You’ll be utilizing this a lot, so it is smart move to memorize it.

Step 3: Apply the zero product rule and work out the linear equation to remove possibilities.

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

Example 1

2x2 + 4x - x2 = 5

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

x2 + 4x - 5 = 0

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




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



We figure out the second-degree equation to get:



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

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

x = 1 x = -5

Next, you have your result! You can check your solution 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

This is it! You've figured out your first quadratic equation utilizing the quadratic formula! Kudos!

Example 2

Let's check out one more example.

3x2 + 13x = 10

Let’s begin, place it in the standard form so it equals 0.

3x2 + 13x - 10 = 0

To figure out this, we will plug in the figures like this:

a = 3

b = 13

c = -10

Work out x using the quadratic formula!



Let’s simplify this as far as feasible by solving it exactly like we executed in the last example. Figure out all easy 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 answer! You can check your work utilizing 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 solve quadratic equations like a pro with a bit of patience and practice!

Given this overview of quadratic equations and their basic formula, children can now tackle this difficult topic with confidence. By opening with this simple explanation, kids acquire a firm grasp before undertaking further complicated theories down in their academics.

Grade Potential Can Guide You with the Quadratic Equation

If you are struggling to get a grasp these theories, you might need a mathematics tutor to assist you. It is best to ask for assistance before you trail behind.

With Grade Potential, you can study all the tips and tricks to ace your next math examination. Become a confident quadratic equation solver so you are ready for the following complicated concepts in your mathematics studies.