Expanding on the topic of algebraic expressions and equations, let's talk about quadratic equations and expressions. For this guide, the term "equation" will be used to refer both expressions and equations where appropriate. A quadratic is an equation with the highest power on an unknown is 2. That is, at least one term in the equation is squared.
Expression: Equation:
The coefficient of the squared term cannot be 0. The other coefficients in the equation can be 0. Consider the following examples:
Quadratic Equation: Quadratic Equation: Not a Quadratic:
Standard Form
A general quadratic equation uses the following format:
In this equation, a represents the coefficient of the squared x term, b represents the coefficient of the non-squared x term, and the letter c represents the constant, or the value without an x attached to it.
If we know that a = 1, b = -3, and c = -10, we can write the following quadratic equation:
We can then use this equation to plug in values of x and solve:
If x = 5: If x = -2
25 - 15 - 10 = 0 4 + 6 - 10 = 0
0 = 0 0 = 0
How can we find the value of x in a quadratic equation?
One approach is using factoring. We can separate the quadratic equation into two quantities that, when multiplied, give us the quadratic equation. I am sure that sounds complicated, but thankfully there is a handy acronym that can help us do this.
To demonstrate this process, let's apply it to the following equation to create a quadratic equation:
FIRST: Multiply the first term in each expression together
OUTER: Multiply the outermost terms in each expression together
INNER: Multiply the innermost terms in each expression together
LAST: Multiply the last term in each expression together
Now we can combine the result of each step back into a single equation. Remember, we write equations in decreasing order of exponents. So, the x-squared term will go first:
Our next step is to combine like-terms:
Now we have a quadratic equation in standard form!
Now that we know how to use FOIL to create a quadratic equation, let's see what it looks like to use FOIL to factor a quadratic. Consider the following quadratic equation:
FIRST: What two values multiply to give you the first term?
Since the first term is just x-squared, the values must both be x
We will actually skip to LAST for the next step. With practice, you'll find you can actually combine this step with the next steps.
LAST: What two numbers multiply to give you the last number?
The last term in our equation is -56, so we can list all possible pairs of values that, when multiplied, give you -56. It is important to consider signs in this step as well.
1 x -56 4 x -14
-1 x 56 -4 x 14
2 x -28 7 x -8
-2 x 28 -7 x 8
INNER/OUTER: Which pair from the previous step adds to give you the middle term?
The middle term in our equation is x, so the sum of the values from the LAST step must equal +1. This leaves only one option: -7, 8 (because -7 + 8 = 1). Plugging those into our factored equation, we get:
Now that our equation is factored, we can set each quantity equal to 0 and solve for x:
x - 7 = 0 x + 8 = 0
x = 7 x = -8
This means that values of 7 or -8 would make our quadratic equation true. That is, if we plug each value in (one at a time) and solve, the quadratic equation should equal 0. Let's check our work:
If x = 7 If x = -8
49 + 7 - 56 = 0 64 - 8 - 56 = 0
0 = 0 0 = 0
When the values of x are not whole numbers, factoring will not work. You can use the quadratic formula to solve any quadratic equation, but it's best when used for complex problems that do not have a factorable solution. Here is the quadratic formula:
Where:
a = the coefficient of the x-squared term
b = the coefficient of the x term
c = the value of the constant
When memorizing this formula, the following phrasing can be helpful,
x equals the opposite of b plus and minus the square root of b-squared minus 4ac all over 2a
Notice it says opposite b instead of negative b. There will be times when b is a negative coefficient. It is easy to think of it as "negative b" and not change the sign on a negative coefficient. Thinking of it as opposite b will help you remember to change the sign on b. Likewise, knowing that it is all over 2a will help you remember that everything that you said first is in the numerator. Finally, you must remember it's plus and minus. This means that you will obtain two answers (one from the addition and one from the subtraction).
Let's work through this with an example.
Step 1: Identify a, b, and c
a = 3 b = 2 c = -7
Step 2: Substitute the values into the formula (plug & chug)
Step 3: Simplify the terms
Step 4: Evaluate the radical term (Can you "pull out" a perfect square?)
Step 5: Simplify the fraction (*all* terms must be divisible to reduce)
All three of the terms are divisible by 2, thus the fraction can be reduced.
At this point, we cannot simplify any further, so the two real numbers that satisfy this equation are:
and 
