Factoring Who Now?

Have you ever tried factoring something and found it impossible? As in, you did all the procedures and found it impossible to factor. Don’t worry; a formula exists for you: the quadratic formula! The quadratic formula looks like this: 👀

How to Use the Quadratic Formula

I know the formula seems like a lot to memorize in the first shot, but you'll get used to it over time with practice. An easy way to remember the formula is to sing it to the tune of “Pop Goes The Weasel.” 🦦

X is equal to negative b / plus or minus the square root / of b squared minus 4 a c / all over 2 a!

Now, let’s look at its various components. The quadratic formula is used to solve trinomial equations. A trinomial equation, a polynomial with three terms, looks like this:

You may have noticed the plus or minus symbol, +. This means that you'll add the equation and subtract it. 

• With the quadratic formula, you'll do the formula twice: 1) adding and 2) subtracting sqrt(b2 - 4ac) to -b. 

• This also means that you'll usually get two answers… which is perfectly fine! 😌

Practice

Now that you know all the components of the quadratic formula, let’s practice! 

Example 1: 

Solve the following equation using the quadratic formula: d^2 + 4d - 12 = 0 

First, plug everything into the formula… 🔌

• a = first term in the trinomial, aka x^2 = 1 

• b = second term, aka x = 4

• c = third term, aka the number = -12

Simplify as many terms as you can! For example, we know that 4 squared is 16 and (-4) x (1) x (-12) = +48 🧩

Stuck? Remember that + means that we'll do addition and subtraction. 🤓

That's right! Your answers will be 2 and -6. 💯

Getting the hang of it? Let’s look at another example! 🧐

Example 2: 

Solve the following equation using the quadratic formula: u^2 - 4u + 8 = 0

As usual, plug in the values for a, b, and c into the formula.

• a = 1

• b = -4

• c = 8

Simplify as much as possible! 🧱

Pause! What do we do if the inside of the square root has a negative number? 😕 

Depending on where you are mathematically, you will either stop here or use something called an imaginary number. Since the square root of a negative number does not exist in the set of real numbers, you can square it with an imaginary number. In this case, sqrt(-16) = 4 x sqrt(-1) = 4i. ☁️

From here, continue simplifying the equation as usual 🧱

Once you can't physically do anything to the answers, you're more than okay to stop at 2 + 2i! 🔨

Example 3: 

Solve the following equation: j^2 - 4j + 4 = 0

You know the drill: plug, plug, plug! ⚡

• a = 1

• b = -4

• c = 4

Simplify as much as possible! 🧱

Since adding or subtracting 0 to anything doesn’t change anything, you don’t have to have to do the plus or minus step! 😈

You will rarely see factors that have one answer, but they DO exist, so don’t be surprised if you see an answer with only one number! Makes life easier for everyone, don't you agree? ☝ 

Let’s spice things up a bit. 🌶️

Example 4: 

Solve the following: -16t^2 - 6t + 11 = 0

The first thing you can do to make things easier is multiply both sides by -1. This minimizes the confusion negative numbers can bring to you, which I’m sure the world can deal less with. 🌎

(-16t^2 - 6t + 11) (-1) = 0 (-1)

16t^2 + 6t - 11 = 0

Now, let's proceed to business as usual! 💼

• a = 16

• b = 6

• c = -11

Simplify as much as possible! 🧱

Simplify the square root. (If you need a refresher on this, check out Fiveable’s math page). ⬛

That 2 in front of the √185 is quite annoying, and it needs to go. In order to get rid of it, divide everything by two! ➗

Now, you can separate the two formulas! 📖

If you, without a calculator, can simplify this, you’re one heck of a genius! Otherwise, this is simplified enough. The way the equations look now is acceptable for an answer 🧠

Example 5:

Solve: 8w^2 = 128

First, divide both sides by 8. 👍

(8w2/8) = (128/8) 

w^2 = 16

Then, move 16 to the other side by subtracting 16 from both sides ➖

w^2 - 16 = 16 - 16 

w^2 - 16 = 0

Now that the equation is fixed, solve it like how we do it normally! 💙

w^2 - 16 = 0

• a = 1

• b = ???

• c = -16

Don’t get confused: b doesn’t exist in this trinomial. In the quadratic formula, you’ll just put 0 since b doesn’t exist ✔️

Simplify as much as possible! 🧱

Now, you can separate the two formulas! 📖

Feeling much more confident? We're at the home stretch! 🏇

Example 6: 

Solve: p^2 - 31 - 2p = -6 - 3p - 3p^2

First, move everything to the left. 👈

p^2 - 31 - 2p + 6 + 3p + 3p^2 = (-6 + 6) (-3p + 3p) (-3p^2 + 3p^2)

p^2 - 31 - 2p + 6 + 3p + 3p^2 = 0

Now, it's time to combine common terms! 🙏

4p^2 + p - 25 = 0

Did you notice that although the terms weren’t in order, they were put in order when simplified? It's important that your expression matches the order of ax^2 + bx + c. Now, we can plug the numbers into the quadratic formula as usual 😁

Simplify as much as possible! 🧱

Now, you can separate the two formulas! 📖

These are your final answers! 📕

Phew! That was a lot, but if you're following along throughout this post, you've essentially mastered the quadratic formula. Give yourself a huge pat on the back! Keep practicing and you'll find quicker and easier ways to plow through these problems 🙌

More Practice

Time to put your quadratic skills to the test! 😏

1. 4h^2 + 8h - 16 = 0

2. d^2 + 17d - 19 = 0

3. x^2 - 81x - 9 = 14x + 23x^2 + 5

4. 125y^2 - 25 = 0

5. 6r^2 + 3r - 36 = 0