November 24, 2022

Quadratic Equation Formula, Examples

If you’re starting to figure out quadratic equations, we are enthusiastic about your venture in math! This is actually where the most interesting things starts!

The information can appear too much at start. But, offer yourself a bit of grace and space so there’s no pressure or strain while working through these questions. To be competent at quadratic equations like an expert, you will need patience, understanding, and a sense of humor.

Now, let’s begin learning!

What Is the Quadratic Equation?

At its center, a quadratic equation is a mathematical equation that states different situations in which the rate of deviation is quadratic or proportional to the square of some variable.

Though it may look like an abstract idea, it is just an algebraic equation expressed like a linear equation. It usually has two answers and utilizes intricate roots to figure out them, one positive root and one negative, through the quadratic equation. Solving both the roots should equal zero.

Definition of a Quadratic Equation

Foremost, keep in mind that a quadratic expression is a polynomial equation that comprises 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 utilize this formula to work out x if we put these terms into the quadratic formula! (We’ll go through it later.)

Ever quadratic equations can be written like this, that results in figuring them out straightforward, relatively 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. Thus, linked to the quadratic formula, we can surely state this is a quadratic equation.

Usually, you can observe these types 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 learned what quadratic equations are and what they appear like, let’s move on to working them out.

How to Figure out a Quadratic Equation Using the Quadratic Formula

Although quadratic equations might look very complex when starting, they can be divided into few easy steps employing an easy formula. The formula for solving quadratic equations consists of setting the equal terms and applying basic algebraic operations like multiplication and division to obtain 2 answers.

After all operations have been performed, we can figure out the values of the variable. The answer take us single step nearer to find result to our actual problem.

Steps to Figuring out a Quadratic Equation Employing the Quadratic Formula

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

ax2 + bx + c=0

Prior to figuring out anything, remember to isolate 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 either side of the equation, total 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 possible

The standard equation you will wind up with should be factored, usually through the perfect square method. If it isn’t workable, plug the variables in the quadratic formula, that will be your best friend for working out quadratic equations. The quadratic formula appears something like this:

x=-bb2-4ac2a

All the terms correspond to the same terms in a conventional form of a quadratic equation. You’ll be employing this a great deal, so it pays to remember it.

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

Now that you possess 2 terms equal to zero, figure out them to attain 2 solutions for x. We have 2 answers due to the fact that the solution for a square root can be both negative or positive.

Example 1

2x2 + 4x - x2 = 5

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

x2 + 4x - 5 = 0

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

a=1

b=4

c=-5

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

x=-bb2-4ac2a

x=-442-(4*1*-5)2*1

We solve the second-degree equation to obtain:

x=-416+202

x=-4362

Next, let’s streamline the square root to obtain 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 work by checking these terms with the initial equation.


12 + (4*1) - 5 = 0

1 + 4 - 5 = 0

Or

-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 check out another example.

3x2 + 13x = 10


Let’s begin, put it in the standard form so it equals zero.


3x2 + 13x - 10 = 0


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

a = 3

b = 13

c = -10


Solve for x using the quadratic formula!

x=-bb2-4ac2a

x=-13132-(4*3x-10)2*3


Let’s streamline this as far as possible by working it out just like we did in the prior example. Figure out all easy equations step by step.


x=-13169-(-120)6

x=-132896


You can work out x by taking 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 through substitution.

3*(2/3)2 + (13*2/3) - 10 = 0

4/3 + 26/3 - 10 = 0

30/3 - 10 = 0

10 - 10 = 0

Or

3*-52 + (13*-5) - 10 = 0

75 - 65 - 10 =0


And this is it! You will solve quadratic equations like nobody’s business with a bit of patience and practice!


Granted this overview of quadratic equations and their basic formula, kids can now go head on against this complex topic with faith. By opening with this simple definitions, children acquire a strong foundation before moving on to further complicated ideas ahead in their studies.

Grade Potential Can Help You with the Quadratic Equation

If you are fighting to get a grasp these concepts, you may require a math tutor to help you. It is best to ask for help before you fall behind.

With Grade Potential, you can study all the helpful hints to ace your next math examination. Turn into a confident quadratic equation problem solver so you are ready for the following big theories in your math studies.