Using the Quadratic Formula to Solve Equations with Literal Coefficients

Using the Quadratic Formula to Solve Equations with Literal Coefficients

Watch this video lesson to find out what steps you need to take to solve equations with literal coefficients using the quadratic formula. Learn why it might be easier to work with literal coefficients than with number coefficients.

Equations with Literal Coefficients

I, personally, like to see and work with equations with literal coefficients. Let me tell you why.

Equations with literal coefficients are your typical equations written with letters instead of numbers. So, instead of seeing an equation like x^2 + 4x + 6 = 0, we would see ax^2 + bx + c = 0.

A literal coefficient is a letter instead of a number. In our example equation, it is the letters a, b, and c. Notice how these letters take the place of the numbers that we are used to seeing.

I actually like to work with these letters because I don't need to go through and do arithmetic calculations. Once I've put in my letters and have simplified my equation as far as I can, I am done. I think this is a lot easier. Keep watching and you can decide for yourself.

The Quadratic Formula

We are going to work with the quadratic formula, which is defined as x = (- b +/- sqrt (b^2 - 4ac)) / 2a. I call formulas like this 'plug-and-play' formulas because you can replace your letters with their appropriate number and then evaluate the formula to find your answer. There are no complicated rules you have to remember other than your order of operations.

One other thing: the quadratic formula is only for solving quadratic equations. Quadratic equations are those equations made up of a polynomial whose degree must be 2. Quadratic equations generally look like our beginning example of x^2 + 4x + 6 = 0. Notice how the degree, or the highest exponent, of the polynomial is a 2.

Now, let's see about working with an equation with literal coefficients instead of numbers. You will see how easy it is.

Standard Form Equation

The first equation with literal coefficients that I want to solve with you is the standard form equation for quadratic equations. It is ax^2 + bx + c = 0. Instead of numbers, we have the letters a, b, and c. What do we know about the same letters in the quadratic formula? They are indeed based off of this standard form equation. Our 'a' is the same 'a' in the quadratic formula, our 'b' is the same 'b,' and our 'c' is the same 'c.' So, for this part, there is not much to do to solve this equation using the quadratic formula. We simply plug in our 'a' for 'a,' our 'b' for 'b,' and our 'c' for 'c,' and we are done. Since the quadratic formula is already at its simplest form, we can't simplify it anymore, and we are left with the formula as it is. That is our answer for solving the standard form equation using the quadratic formula.

x = (- b +/- sqrt (b^2 - 4ac)) / 2a

Even though the answer looks complicated, it is at its simplest form. I can't reduce it any further so I have nothing to worry about. I am totally done. No arithmetic to do, no adding, subtracting, multiplying, or dividing. I am totally done. See how easy it is?

Random Quadratic Equations

It gets even easier with random quadratic equations. We only have three choices for our random quadratic equations. They are: ax^2 + bx = 0, ax^2 + c = 0, and ax^2 = 0. I will show you how to solve one of them, and you can follow the same pattern to solve the others on your own.

The one we are going to solve together is ax^2 + c = 0. What I see here is that I have 'a' and 'c,' but no 'b.' Since I have no 'b,' that means my 'b' equals zero. So, going to the quadratic formula, I will put in 'a' for 'a,' 0 for 'b,' and 'c' for 'c.'

x = (- 0 +/- sqrt (0^2 - 4ac)) / 2a

I simplify this, and I get:

x = (sqrt (- 4ac)) / 2a

At this point, I can't simplify any further. That means I am done, and this is my answer.

I think this is a lot easier than doing it with numbers because, again, I didn't have to do any arithmetic. I just plugged in my letters and reduced as far as I can.

To solve the other equations, you would follow the same steps and for each letter that is missing, you would plug in a zero into the quadratic. You would then simplify to get your answer.

Lesson Summary

In summary, a literal coefficient is a letter instead of a number, and an equation with literal coefficients is an equation written with letters instead of numbers. To solve quadratic equations with literal coefficients, you plug in your letters into their respective slots in the quadratic formula, simplify, and then you are done. If your equation is missing a letter, you plug in a zero for it before simplifying.