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Quadratic Functions and their Relationship to Quadratic Equations

Quadratic functions are functions that can be written in the form $\,y = ax^2 + bx + c\,$ 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.

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:

$\displaystyle ax^2 + bx + c = 0$	original equation
$\displaystyle ax^2 + bx = -c$	subtract $\,c\,$ from both sides
$\displaystyle x^2 + \frac{b}{a}x = -\frac{c}{a}$	divide both sides by $\,a\ne 0\,$; the technique of completing the square only works when the coefficient of the squared term is $\,1\,$
$\displaystyle x^2 + \frac{b}{a}x + {(\frac{b}{2a})}^2 = -\frac{c}{a} + \frac{b^2}{4a^2}$	notice that $\,\cssId{s79}{\frac{b/a}{2}} \cssId{s80}{= \frac{b}{a}\div 2} \cssId{s81}{= \frac{b}{a}\cdot \frac{1}{2}} \cssId{s82}{= \frac{b}{2a}}\,$; add $\,{(\frac{b}{2a})}^2 = \frac{b^2}{4a^2}\,$ to both sides; this is the correct number to add to turn the left side into a perfect square
$\displaystyle{\left(x + \frac{b}{2a}\right)}^2 = -\frac{c}{a}\cdot \frac{4a}{4a} + \frac{b^2}{4a^2}$	rewrite left side as a perfect square; get a common denominator on the right
$\displaystyle{\left(x + \frac{b}{2a}\right)}^2 = \frac{b^2 - 4ac}{4a^2}$	add fractions, simplify
$\displaystyle x + \frac{b}{2a} = \pm\sqrt{\frac{b^2 - 4ac}{4a^2}}$	the equation $\,z^2 = k\,$ has solutions $\,z=\pm\sqrt{k}\ $; recall that $\,z=\pm\sqrt{k}\,$ is just a convenient shorthand for $\,(z = \sqrt{k}\ \ \text{or}\ \ z = -\sqrt{k})$
$\displaystyle x + \frac{b}{2a} = \pm \frac{\sqrt{b^2-4ac}}{\sqrt{4a^2}}$	use the rule $\displaystyle \,\sqrt{\frac{A}{B}} = \frac{\sqrt{A}}{\sqrt{B}}\,$
$\displaystyle x + \frac{b}{2a} = \pm \frac{\sqrt{b^2-4ac}}{\sqrt{4}\sqrt{a^2}}$	use the rule $\displaystyle \,\sqrt{AB} = \sqrt{A}\cdot\sqrt{B}\,$
$\displaystyle x + \frac{b}{2a} = \pm \frac{\sqrt{b^2-4ac}}{2|a|}$	use the rule $\,\sqrt{A^2} = |A|\,$

Notice that if $\,a\gt 0\,$, then $\,|a|=a\,$, so the right side of the previous equation becomes $\,\pm\frac{\sqrt{b^2-4ac}}{2a}\,$.

Also, if $\,a\lt 0\,$, then $\,|a|=-a\,$, so the right side becomes $\,\pm\frac{\sqrt{b^2-4ac}}{2(-a)} = \mp\frac{\sqrt{b^2-4ac}}{2a}\,$.

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

$\displaystyle x + \frac{b}{2a} = \pm \frac{\sqrt{b^2-4ac}}{2a}$	the plus/minus allows elimination of the absolute value
$\displaystyle x=-\frac{b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a}$	add $\displaystyle\,-\frac{b}{2a}\,$ to both sides
$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$	add fractions

In summary, we have:

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:

• If $\,b^2 - 4ac\gt 0\,$, then $\sqrt{b^2-4ac}\,$ is a positive real number.
  In this case,   $\displaystyle\,\frac{-b + \sqrt{b^2-4ac}}{2a}\,$   and   $\displaystyle\,\frac{-b - \sqrt{b^2-4ac}}{2a}\,$   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.
• If $\,b^2 - 4ac= 0\,$, then $\sqrt{b^2-4ac} = 0\,$.
  In this case,   $\displaystyle\,\frac{-b + \sqrt{b^2-4ac}}{2a} = \frac{-b+0}{2a}\,$   and   $\displaystyle\,\frac{-b - \sqrt{b^2-4ac}}{2a} = \frac{-b-0}{2a}\,$   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.
• 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.

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:

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Complex Numbers