November 24, 2022

Quadratic Equation Formula, Examples

If this is your first try to solve quadratic equations, we are enthusiastic regarding your adventure in math! This is really where the fun starts!

The data can look enormous at start. Despite that, provide yourself a bit of grace and space so there’s no rush or strain while working through these problems. To be competent at quadratic equations like a professional, you will need a good sense of humor, patience, and good understanding.

Now, let’s begin learning!

What Is the Quadratic Equation?

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

Though it seems like an abstract concept, it is simply an algebraic equation expressed like a linear equation. It ordinarily has two solutions and uses complicated roots to work out them, one positive root and one negative, employing the quadratic equation. Unraveling both the roots the answer to which will be zero.

Meaning of a Quadratic Equation

First, 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 usual form is:

ax2 + bx + c

Where “a,” “b,” and “c” are variables. We can use this equation to work out x if we put these variables into the quadratic formula! (We’ll go through it later.)

Any quadratic equations can be written like this, which makes solving them straightforward, relatively speaking.

Example of a quadratic equation

Let’s compare the given equation to the last equation:

x2 + 5x + 6 = 0

As we can see, there are two variables and an independent term, and one of the variables is squared. Thus, linked to the quadratic equation, we can surely tell this is a quadratic equation.

Commonly, you can see these kinds of formulas when measuring a parabola, that is a U-shaped curve that can be graphed on an XY axis with the data that a quadratic equation gives us.

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

How to Solve a Quadratic Equation Utilizing the Quadratic Formula

While quadratic equations might seem very complicated when starting, they can be broken down into multiple simple steps using a straightforward formula. The formula for working out quadratic equations involves setting the equal terms and utilizing fundamental algebraic operations like multiplication and division to achieve two solutions.

After all operations have been executed, we can figure out the numbers of the variable. The solution take us single step nearer to work out the solutions to our actual problem.

Steps to Solving a Quadratic Equation Utilizing the Quadratic Formula

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

ax2 + bx + c=0

Ahead of solving anything, remember to isolate the variables on one side of the equation. Here are the three steps to figuring out a quadratic equation.

Step 1: Write the equation in conventional mode.

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

Step 2: Factor the equation if workable

The standard equation you will conclude with must be factored, ordinarily through the perfect square method. If it isn’t feasible, plug the variables in the quadratic formula, which will be your best friend for working out quadratic equations. The quadratic formula seems something like this:

x=-bb2-4ac2a

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

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

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

Example 1

2x2 + 4x - x2 = 5

Now, let’s piece down this equation. Primarily, streamline and put it in the standard form.

x2 + 4x - 5 = 0

Immediately, let's identify 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 work out quadratic equations, let's replace this into the quadratic formula and solve for “+/-” to involve each square root.

x=-bb2-4ac2a

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

We work on the second-degree equation to get:

x=-416+202

x=-4362

Now, let’s streamline the square root to attain 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 work by checking these terms with the first equation.


12 + (4*1) - 5 = 0

1 + 4 - 5 = 0

Or

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

25 - 20 - 5 = 0

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

Example 2

Let's try another example.

3x2 + 13x = 10


Initially, put it in the standard form so it results in 0.


3x2 + 13x - 10 = 0


To solve this, we will substitute in the numbers 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 simplify this as far as feasible by solving it just like we executed in the previous example. Solve all easy equations step by step.


x=-13169-(-120)6

x=-132896


You can work out 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 result! You can revise your workings 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 that's it! You will work out quadratic equations like a professional with a bit of patience and practice!


Given this overview of quadratic equations and their rudimental formula, kids can now take on this challenging topic with faith. By starting with this straightforward definitions, children secure a solid grasp before taking on more intricate ideas down in their studies.

Grade Potential Can Assist You with the Quadratic Equation

If you are struggling to understand these theories, you may need a mathematics tutor to help you. It is better to ask for guidance before you get behind.

With Grade Potential, you can understand all the handy tricks to ace your subsequent math examination. Turn into a confident quadratic equation solver so you are prepared for the following complicated concepts in your mathematics studies.