November 24, 2022

Quadratic Equation Formula, Examples

If you going to try to solve quadratic equations, we are thrilled about your journey in math! This is indeed where the fun starts!

The data can look enormous at start. However, provide yourself some grace and room so there’s no pressure or strain while working through these problems. To be efficient at quadratic equations like a pro, you will require patience, understanding, and a sense of humor.

Now, let’s begin learning!

What Is the Quadratic Equation?

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

Though it might appear similar to an abstract idea, it is simply an algebraic equation expressed like a linear equation. It ordinarily has two answers and uses intricate roots to solve them, one positive root and one negative, using the quadratic equation. Unraveling both the roots the answer to which will be zero.

Meaning of a Quadratic Equation

Foremost, keep in mind that a quadratic expression is a polynomial equation that includes 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 figure out x if we replace these numbers into the quadratic equation! (We’ll get to that later.)

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

Example of a quadratic equation

Let’s compare 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, compared to the quadratic formula, we can confidently state this is a quadratic equation.

Commonly, you can see these types of equations when scaling a parabola, which is a U-shaped curve that can be graphed on an XY axis with the details that a quadratic equation gives us.

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

How to Figure out a Quadratic Equation Utilizing the Quadratic Formula

Although quadratic equations may appear greatly complex when starting, they can be cut down into several easy steps employing an easy formula. The formula for figuring out quadratic equations involves creating the equal terms and using rudimental algebraic operations like multiplication and division to obtain two answers.

Once all functions have been executed, we can work out the values of the variable. The results take us one step closer to discover answer to our actual problem.

Steps to Solving a Quadratic Equation Employing the Quadratic Formula

Let’s promptly place in the general quadratic equation once more so we don’t omit what it looks like

ax2 + bx + c=0

Prior to solving anything, remember to separate the variables on one side of the equation. Here are the three steps to work on a quadratic equation.

Step 1: Note the equation in standard mode.

If there are variables on both sides of the equation, sum all similar terms on one side, so the left-hand side of the equation equals zero, just like the standard mode of a quadratic equation.

Step 2: Factor the equation if feasible

The standard equation you will wind up with must be factored, ordinarily utilizing the perfect square method. If it isn’t possible, plug the terms in the quadratic formula, that will be your closest friend for figuring out quadratic equations. The quadratic formula looks similar to this:

x=-bb2-4ac2a

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

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

Now once you have two terms equal to zero, figure out them to obtain two answers for x. We have two answers because the answer for a square root can be both negative or positive.

Example 1

2x2 + 4x - x2 = 5

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

x2 + 4x - 5 = 0

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

a=1

b=4

c=-5

To work out quadratic equations, let's put this into the quadratic formula and solve for “+/-” to include each square root.

x=-bb2-4ac2a

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

We solve the second-degree equation to achieve:

x=-416+202

x=-4362

Now, let’s streamline the square root to get two linear equations and solve:

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

x = 1 x = -5


After that, you have your solution! You can revise your work by using 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 worked out your first quadratic equation utilizing the quadratic formula! Kudos!

Example 2

Let's work on another 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 put 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 clarify this as much as feasible by figuring it out just like we performed in the last example. Work out all simple equations step by step.


x=-13169-(-120)6

x=-132896


You can figure 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 result! You can check 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

Or

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

75 - 65 - 10 =0


And that's it! You will figure out quadratic equations like nobody’s business with little practice and patience!


Given this summary of quadratic equations and their basic formula, children can now go head on against this complex topic with assurance. By opening with this easy explanation, children secure a solid foundation before moving on to more complicated ideas later in their studies.

Grade Potential Can Help You with the Quadratic Equation

If you are struggling to understand these concepts, you may need a mathematics instructor to assist you. It is better to ask for assistance before you get behind.

With Grade Potential, you can understand all the handy tricks to ace your next mathematics exam. Turn into a confident quadratic equation solver so you are ready for the ensuing intricate concepts in your mathematical studies.