How to Identify Lead Coefficients in Quadratic Equations

Instructor: Stephanie Matalone

Stephanie taught high school science and math and has a Master's Degree in Secondary Education.

In this lesson, will will go over some basics of quadratic equations and coefficients, specifically leading coefficients and how to find them. We will review examples and special cases.


What's a coefficient again? Before we even talk about leading coefficients and quadratic equations, let's go over what a simple coefficient is. A coefficient is the number being multiplied by a variable. Variables are simply the letters we see in math that stand for unknown values. You often see x and y as variables.

So if we have the term 6x, the coefficent would just be 6. The number 6 is being multiplied by the variable x which makes it the coefficient. Now let's say we have another term 129xyz. 129 would be the coefficient, even though there are three variables in that term. If we have the expression 12x + 3y, both 12 and 3 would both be coefficients.


Leading Coefficients

So we understand coefficients, but what exactly is a leading coefficient? Think about the term... leading usually means you are out in front or first! When an expression or equation is written in the correct form, that is, in descending order of exponents, the leading coefficent will come first. This means, the leading coefficent is the coefficient attached to the variable with the highest exponent.

Thus, when your expression is written with the highest exponent terms first, the leading exponent will but just that... in the lead and out in front! But, sometimes your expressions or equations will not be written in descending order, so be careful!

Let's look at an example. Let's say we have the expression 7y 9 + 3y 5 + 2y 2 This is written in the correct order with the term with the highest exponent of 9 coming first, followed by the next highest exponent of 5 coming next and the smallest exponent of 2 coming last. Since it is in the correct order, the leading coefficent is simply the coefficient that comes first. In this case, that would be 7! Another way to look at it is that the highest exponent on a variable in this expression is 9. What is the coefficent for that term? It's 7!

Leading Coefficient

In another expression -3x 2 + 4x 5 , the terms are not ordered correctly. To find the leading coefficient here, we can either order it correctly as 4x 5 - 3x 2 and find the leading coefficient that way. Or we can simply look for the term with the highest exponent and find its coefficent. In this case, the highest exponent is 5, thus the leading coefficient will be -3.

Quadratic Equations

Now that we got all that under our belts, let's talk quadratic equations. Quadratic equations are equations in which the highest exponent on a variable is a two. They are typically in the form of y = ax 2 + bx + c where a cannot equal zero. An example of a quadratic equation is y = 4x 2 + 2x + 1 where a = 4, b = 2, and c = 1.

By definition, we know quadratic equations will have a term with an exponent of two and that will be the highest exponent. This tells you that the leading coefficient will always be attached to a variable with an exponent of two! Looking at the general form, y = ax 2 + bx + c, the leading coefficient will always be the a value.

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