Introduction to Quadratic Equations
In this section you’ll learn the basics. But, even though these are the basics doesn’t mean that there is not a lot of information, or that it is all simple. Give yourself time to work on this, take good notes, watch the video and try the practice problems posted below, and listed in the video.
Use the tabs below to navigate through the notes, practice problems and video. When you’ve mastered this level, move onto the next page using the navigation tabs above.
A quadratic equation is a polynomial equation with a single input and single output that also has a degree of two. Clear as mud, right? Let’s break that down a bit, to make it concrete.
- A quadratic equation is an equation.
- it is equal to something.
- A quadratic equation is a polynomial equation.
- polynomials (basically) have Natural Number exponents and Real coefficients.
- A quadratic equation has a degree of 2.
- Basically the largest exponent is two.
- The input is x, the output is
Below are a few examples of quadratic equations and functions. Note: All quadratic equations we deal with are functions because for each input there is exactly one output.
The topic of quadratics, if it can be called such, is pretty nuanced and vast. There are a lot of minor details that accumulate into big issues for students. We will introduce a few here that will be important throughout, or that can creep up unexpectedly. It is easy to be overwhelmed and to have some small detail slip from your awareness, only to cause distress and confusion in the future. Let’s try to mitigate that as much as possible in this introductory section.
Linear v Quadratic
Let’s compare quadratic equations to linear equations. A linear equation has a degree of one, with an input and output, one dependent (y), and the other independent (x). Not all linear equations are functions, the exception being the case where the coefficient of y is zero, as in x = 3. This is a vertical line, and will have one x – intercept and no y – intercepts. Aside from horizontal and vertical lines, linear equations have one x – intercept (and that is found by replacing y with zero), and one y – intercept (found by replacing x with zero). The graph of a linear equation will pass through, at most, three quadrants.
What makes linear equations linear is that the rate of change is constant, they have a constant slope. This makes the solutions (when graphed) co-linear, forming a straight line.
A quadratic equation can have more than one x – intercept. It can have two, one, or zero. It will always have one y – intercept, and also has a very important feature that linear equations do not have: a vertex.
The shape of a quadratic equation, when graphed, is called a parabola. They open up, or down, but a parabola from a quadratic function never opens sideways.
Quadratic equations do have slopes, plural. Each point on the graph of a quadratic equation has a different slope. The slope is NOT constant, and is not a key feature we seek.
Summary: A linear equation is a line. The slope is constant. A linear equation has at most one x – intercept and one y – intercept. A quadratic equation does NOT have a constant slope, the shape is called a parabola, and it has zero, one or two x – intercepts, and always one y – intercept. A key feature of a parabola that a linear equation does not have is called a vertex. The vertex is the minimum or maximum value of the equation. (More on the vertex later.)
The standard form of a quadratic equation (or function) is a shape, or pattern, we use to write many quadratic equations. It is organized and simple, and it allows us to reference coefficients easily. Coefficients are the numbers multiplying with the variable. Standard form is as follows.
Here, a is the coefficient of x2, and b the coefficient of x, and c is called the constant term as its value will not change with different values of x. An example of a quadratic equation in standard form is Here a = 1, b = 2, and c = -6.
A common source of confusion occurs when b = 0, as in y = 9x2 – 25. The b is NOT the second number you see, it is the coefficient of x. There is another common way of writing quadratic equations called Vertex Form, but we will get into that in the future.
Building Quadratic Equations
Not all quadratic equations can be written as the product of two binomials, but many are. Initially, those will be the quadratic equations we deal with. For example, if you had the two binomials x – 5 and 2x + 3, and you multiplied them, the product would be quadratic.
If this was written as a function, we would have the following.
The reason we spent so much time and energy factoring these quadratic expressions in the polynomials section is because factoring is the simplest way to find the x – intercepts of quadratic equations. Note: x – intercepts are also called solutions, roots, and zeros.
The number 25 is a square number because it is the product of another number that has been squared (5). Polynomials are numbers. The binomial x + 5 can be squared. (x + 5)2 is (x + 5)(x + 5), which has the product of x2 + 10x + 25. Later in the unit on quadratics perfect squares will play a key role.
There’s a relationship in quadratics that are perfect squares. Let’s look at a few examples where the leading coefficient (a = 0, in ax2 + bx + c).
In each of these there is a relationship between b and c. Do you see what it is? Look again, if you haven’t seen it yet? If you find the pattern on your own, you’ll be more likely to remember. In the future we will be taking non-factorable quadratic equations and making them perfect squares by manipulating the numbers algebraically.
In a perfect square, when a = 1 (the leading coefficient is 1), half of b, squared, is c. The constant term is always positive because it is a square, and a negative number squared is positive.
A quadratic equation will always have one y – intercept, which can be found by replacing x with zero, and solving for y. In function notation, this is f (0).
In standard form, the y – intercept can be found by inspection, just looking. When you replace x with zero, all that remains is the constant term. So the y – intercept is (0, c), when you have f(x) = ax2 + bx + c.
But, when in vertex form, f(x) = a(x – h)2 + k, you must use the order of operations to find the y value of the y – intercept. Still, just replace x with zero, then preform the arithmetic to find the y – intercept.
The vertex is perhaps the most important feature of a graphed quadratic equation. There is a vertical line of symmetry that passes through the x – coordinate of the vertex. The vertex is also the minimum output value (y) or the maximum output value (y). If the parabola goes up, then a is positive, and the vertex is the minimum. If a is negative, then the parabola is what we call going down, and the y of the vertex will be the maximum output of the equation.
The vertex can be found in two ways. From standard form we can use a formula, which is below.
This formula looks confusing, but let’s take a second to understand it. The formula is written as a coordinate, (x, y). The x – coordinate is found by taking the opposite sign of b, and dividing it by twice of a. The y – coordinate is found by taking the x – value and plugging it into the equation. Let’s see an example.
Here a = 1, b = -2, and c = 6. The formula for the x – coordinate is where we begin.
The line (axis) of symmetry for the graph will be x = 1. To find the y – coordinate of the vertex we simply plug this value into the original. So we evaluate f(1).
The vertex is then (1, 5), which is in the first quadrant. Since a is positive, this graph will go “up.” Since the vertex is the minimum value for a quadratic equation that is going up, and that minimum value is above the x – axis, this graph will have zero Real x – intercepts.
Let’s see another, more complicated, example of finding the vertex, one where the x – coordinate doesn’t reduce to an integer value.
Here a = 3 and b = 2. Let’s plug those value into the formula for the x – coordinate.
To find the y – coordinate, plug this value of x into the equation. While it will look complicated, there’s a pattern that we can make use of to help with the calculations.
Be careful to follow the order of operations here…exponents before reducing!
Now reduce, if possible. For the sake of ease, do NOT combine the last two numbers, yet. You’ll see why soon.
Notice now the second number is twice of the first (ignoring the sign … the magnitude is twice as large). This is a great help for calculation, though we might not need the clue here. But A – 2A = – A, and 5 – 10 = – 5, and 40 – 80 = -40, and
Note: It is best to do arithmetic with improper, but reduced fractions. You would not want to plug in the value because is easier to perform calculations with. However, for your final answer, this is perfectly appropriate. Also, when finding the vertex, use fractions, not decimals. Decimals are approximations, which will create errors by compounding the approximation.
We will spend more time on the vertex in the future.
Note: The vertex and the y – intercept can be the same value. This happens when b = 0.
The x – intercepts are where the parabola crosses, or touches, the x – axis. There can be zero, one or two x – intercepts. There are always two solutions to quadratic equations that are set equal to zero, but sometimes the solutions are repeated (x2 = 4), and sometimes they’re Imaginary numbers, as in x2 = – 4.
To begin, the way you’ll find the x – intercepts is by factoring and using a property of zero. If two numbers are multiplied and the product is zero, then one, or both of the numbers must be zero.
A × B = 0
Either A = 0, or B = 0, or both equal zero.
A binomial is number with two terms, like x – 2. Another example is 3x + 5. If we multiplied these two binomials and the product was zero, it would look like this below.
(x – 2)(3x + 5) = 0
If you were asked to find the values of x that make the equation true (AKA solve for x), you could set each factor equal to zero, and solve each equation.
x – 2 = 0 OR 3x + 5 = 0
Using inverse operations (SADMEP), we’d arrive at two solutions. Either x = 2, or x = -5/3.
Suppose you had to find the x – intercepts for y = 3x2 – 7x – 10. The way you find x – intercepts is to replace y with zero and solve for x. If the quadratic equation is factorable, try factoring. That’s perhaps the simplest way to find the x – intercepts. The factors of this equation are (x – 2)(3x + 5).
Note: If the value of b = 0, then you can use inverse operations to find the x – intercepts. However, if b is NOT zero, you cannot isolate the variable x with inverse operations because you have an x and an x2.
Let’s see an example where we find the intercepts (both x and y), and the vertex.
To start with, let’s find the easiest thing, the y – intercept. To find the y – intercept, replace x with zero.
The vertex is found using the formula:
Let’s start with the x – coordinate.
Now, plug in that value to find y:
Now, for the x – intercepts. Set the equation equal to zero (f(x) = 0), and solve for x. Factoring is the best way to start, though you’ll learn what to do when factoring fails in the future.
0 = 2x2 – 4x + 2
Here we need to find the value(s) of x that would satisfy the equation. If x = 1, then we would have a true statement. We only have one x – intercept here, (1, 0), which is the same as the vertex.
Here’s what we found.
Vertex: (1, 0), y – intercept: (0, 2), x – intercepts: (1, 0)
In future sections we will explore, in greater detail, how to sketch the graphs of quadratic equations and the nuances of finding the key features of the graphs.