Lesson Who is who in quadratic equations

Lessons PROOF of quadratic formula... and Introduction into quadratic equations of this module explain what is the quadratic formula and how to use it to solve quadratic equations.
So, I suppose you know these issues.
You will know more and understand it better after reading this lesson.

Let's consider quadratic equation

.                                         (1)

Figure 1 below shows the typical plot of the quadratic function, which is left side of this equation. I guess you saw such plots many times. The curve in Figure 1 is called parabola.


Figure 1

Let's perform an operation of square completing to get

(2)
For what value of x the quadratic function (1), (2) has the minimum/maximum?
The answer is: for .
To check it, simply substitute this value of x into quadratic function (2),
and you will see that the term is zero for this value of x.
It is also easy to see that vertical line is the symmetry line
of the parabola. Figure (2) below illustrates this fact by showing the symmetry line colored in green.


Figure 2

Next question: what is the value of the quadratic function (1), (2) at ?
In order to answer this question, simply substitute into function (2),
and you will get the value . This value is marked by blue horizontal
line in Figure 3 below.


Figure 3

Last question: What is the distance along the X-axis from the green vertical symmetry line
to the root(s) of the quadratic function? The answer is , exactly the value produced by the expression of the standard quadratic formula after sign(s) +/-.