# The Quadratic Formula

You may want to review preliminary information about quadratic functions and related concepts:

- Parabolas
- Equations of Simple Parabolas
- Quadratic Functions and the Completing the Square Technique
- Algebraic Definition of Absolute Value
- Solving More Complicated Equations Involving Perfect Squares

A *quadratic equation*
is an equation of the form $\,ax^2 + bx + c = 0\,,$
where $\,a\ne 0\,.$

The *quadratic formula*
solves quadratic equations.
Just read off $\,a\,,$
$\,b\,,$ and $\,c\,,$
and substitute into the quadratic formula:

The Quadratic Formula: $$\cssId{s7}{x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}}\ \ \text{(for $a\ne 0$)}$$

For example, solving the quadratic equation ‘$\,14x + 3x^2 = 5\,$’ is as simple as this:

$\color{blue}{b = 14}$

$\color{green}{c = -5}$

- $a\,$ (the coefficient of the $\,x^2\,$ term)
- $b\,$ (the coefficient of the $\,x\,$ term)
- $c\,$ (the constant term)

or

$\displaystyle \cssId{s32}{x} \cssId{s33}{= \frac{-14\,\color{red}{\large{\mathbf{-}}}\,16}{6}} \cssId{s34}{= \frac{-30}{6}} \cssId{s35}{= -5}$

Write the ‘$\,\color{orange}{\large{\mathbf{+}}}\,$’ and ‘$\,\color{red}{\large{\mathbf{-}}}\,$’ solutions separately.

*sure*they're equal. Voila!

## Quadratic Functions and their Relationship to Quadratic Equations

Quadratic functions are functions that can be written in the form $\,ax^2 + bx + c\,$ for $\,a\ne 0\,.$ Every quadratic function graphs as a parabola with directrix parallel to the $x$-axis.

The graphs of quadratic functions can have three different $x$-intercept situations, as shown below:

- no $x$-intercepts
- exactly one $x$-intercept
- two different $x$-intercepts

Points on the $x$-axis have their $y$-value equal to zero. Thus, to find the $x$-intercepts for any curve, you set $\,y\,$ equal to zero and solve for $\,x\,.$

In particular, to find the
$x$-intercepts
of a quadratic function
$\,y = ax^2 + bx + c\,$
($\,a\ne 0\,$),
it is necessary to solve the equation
$\,ax^2 + bx + c = 0\,$
(which is called a *quadratic equation*).

The formula that gives the solutions
to this equation is called *the quadratic formula*,
and is derived next.

## Derivation of the Quadratic Formula

### Solving the Equation $\,ax^2 + bx + c = 0\,,$ where $\,a\ne 0\,$

Using the technique of completing the square, the equation is transformed to the form $\,z^2 = k\,,$ as follows:

$$z = \sqrt{k}\ \ \text{or}\ \ z = -\sqrt{k}$$

In both cases, the same two values for the right side result. Thus, we continue with a simplified right side:

In summary, we have:

Let $\,a\ne 0\,.$

The solutions to the equation $\,ax^2 + bx + c = 0\,$ are given by:

$$ \cssId{s117}{x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}} $$## The Discriminant of a Quadratic Equation or Quadratic Function

The expression $\,b^2 - 4ac\,$ that appears under the square root in the quadratic formula is critical in determining the nature of the solutions to the quadratic equation $\,ax^2 + bx + c = 0\,,$ as follows:

### Positive Discriminant

If $\,b^2 - 4ac\gt 0\,,$ then $\sqrt{b^2-4ac}\,$ is a positive real number.

In this case,
$$\cssId{s123}{\frac{-b \color{red}{+ \sqrt{b^2-4ac}}}{2a}}$$
and
$$\cssId{s124}{\frac{-b \color{red}{- \sqrt{b^2-4ac}}}{2a}}$$
are *different* real numbers.
Thus, there are two different real number
solutions to the quadratic equation $\,ax^2 + bx + c = 0\,,$
and the graph of the quadratic function
$\,ax^2+bx+c\,$ has two different
$x$-intercepts.

### Discriminant Equal to Zero

If $\,b^2 - 4ac= 0\,,$ then $\sqrt{b^2-4ac} = 0\,.$

In this case,
$$\cssId{s131}{\frac{-b + \sqrt{b^2-4ac}}{2a} = \frac{-b\color{red}{+0}}{2a}}$$
and
$$\cssId{s132}{\frac{-b - \sqrt{b^2-4ac}}{2a} = \frac{-b\color{red}{-0}}{2a}}$$
are *the same numbers*.

Thus, there is exactly one real number solution to the quadratic equation $\,ax^2 + bx + c = 0\,,$ and the graph of the quadratic function $\,ax^2 + bx + c\,$ has only one $x$-intercept.

### Negative Discriminant

If $\,b^2 - 4ac\lt 0\,,$ then $\,\sqrt{b^2-4ac}\,$ is not a real number.

In this case, there are no real number solutions to the quadratic equation $\,ax^2 + bx + c = 0\,,$ and the graph of the quadratic function $\,ax^2+bx+c\,$ has no $x$-intercepts.

### The Discriminant Discriminates...

Thus, the expression
$\,b^2 - 4ac\,$ helps us to
*discriminate* between
the various types of solutions to a quadratic equation,
and the various
$x$-intercept situations for
a quadratic function.

Thus, we have the following definition:

Let $\,a\ne 0\,.$

The expression
$\,b^2 - 4ac\,$ is called
the *discriminant*
of the quadratic equation
$\,ax^2 + bx + c = 0\,.$

Similarly, the expression $\,b^2 - 4ac\,$ is called
the *discriminant*
of the quadratic function $\,ax^2 + bx + c\,.$