Quadratic Functions

Definition of a Quadratic Function

A quadratic function can be written in the form of $ax^2+bx+c$ where $a \neq 0$. If $a = 0$ then the function would be linear and not quadratic!


Roots and Zeros are the values of the independent variable (often x) that make the entire function zero. These coincide with the x-intercepts. The only difference being that the x-intercepts are coordinate points on a graph, roots or zeros are a value of a variable.

Standard Form, Vertex Form and Factored Form Quadratics are often written in different forms. Each has is own usefulness. For more on quadratic forms try this page.

Finding Zeros

A huge number of problems involving quadratics involve finding the zeros. There are three main ways of finding zeros:

Quadratic Formula

The quadratic formula is in your IB Formula Booklet. I'll put it here for reference too:

\begin{align} x=\frac{b \pm \sqrt{b^2-4ac}}{2a} \end{align}

This is a pretty simply "plug-and-chug" equation. The letters in the formula correspond to the quadratic form $y=ax^2+bx+c$. Its key to remember that in general two solutions come from the Quadratic Formula… this is due to the $\pm$ in the equation.

The number of zeros and the type of zeros can be determined from the discrimnate:

\begin{align} \Delta = b^2 - 4ac \end{align}

This is nothing more than the part of the quadratic equation that is in the square root. If the discriminate is zero the there will be only one solution. There are still two zeros, but they are repeated. This also coincides with the vertex being on the x-axis! If the discriminate is greater than zero there will be two solutions and they will both be real. If the discriminate is negative there will be two solution but both zeros with be imaginary.

Discriminate Solutions & Zeros
$\Delta = 0$ 1 solution, two repeated real zeros
$\Delta > 0$ 2 solutions, two distinct real zeros
$\Delta < 0$ 2 solutions, two distinct imaginary zeros

GDC - Graphing Display Calculator

Your calculator has a function for finding the zero of any function. This can be found under the "calc" option at the top right (Ti-84).

Look here for screen shots of exactly how to do it. You will need to scroll down a bit to find the part on zeros.


This third option is a good option if you are "good" at factoring and can do it quickly. Otherwise the above two methods are more "sure-fire." Factoring can be a waste of time as not all quadratics can be factored, at least not with real numbers…

Factoring for those unfamiliar with it looks like:

\begin{equation} x^2+bx+c=(x+d)(x+e) \end{equation}

Where $d+e=b$ and $(d)(e)=c$.

For those who really like to factor or are faced with a question that requires them to factor see the page regarding the Factor Theorem.

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