1.  A polynomial function of the 2nd degree has what form?

y = ax² + bx + c

2.  A quadratic equation has what form?

ax² + bx + c = 0

3.  What do we mean by a root of a quadratic?

A solution to the quadratic equation.

4.  A quadratic always has how many roots?

5.  The graph of a quadratic is always the form called -- ?

A parabola.

6.  What are the three methods for solving a quadratic equation,
6.   that is, for finding the roots?

1.  Factoring.   2.  Completing the square.

3.  The quadratic formula.

7.  If a product of factors is 0 -- if ab = 0 -- then what can you
7.   conclude about the factors a, b ?

Either a = 0, or b = 0.

Example 1.   Solution by factoring.   f(x) = x² −2x −3.  Find the roots of f(x), and sketch the graph of y = f(x).

 Solution.   x² −2x −3 = (x + 1)(x − 3).

Example 2.  A double root.   f(x) = x² −10x + 25.  Find the roots of
f(x), and sketch the graph of y = f(x).

 Solution.   x² −10x + 25 = (x − 5)(x − 5) = (x − 5)².  The "two" roots are 5, 5.   5 is called a double root.  At a double root, the graph does not cross the x-axis.  It just touches it.

A double root occurs when the quadratic is a perfect square trinomial:  x² ±2ax + a²; that is, when it is the square of a binomial: (x ± a)².

Example 3.   How many real roots, i.e. roots that are real numbers, has the quadratic of each graph?

 Answer.   Graph a) has two real roots.  It has two x-intercepts.

Graph b) has no real roots.  It has no x-intercepts.  Both roots are complex.

Graph c) has two real roots.  But they are a double root.

Example 4.  Quadratic inequality.   Inspect this graph of

y = x² − 4x − 5

to solve this inequality:

x² − 4x − 5 < 0

 Solution.   For which values of x will that quadratic be negative?  That is, where is the graph below the x-axis?

The graph is negative between the roots, which are −1 and 5.  The solution to the inequality is  −1 < x < 5.

We can also observe that the quadratic will have positive values -- the graph will be above the x-axis -- to the left and right of the roots:

x < −1  or  x > 5.

While the quadratic will have the value 0 at the roots.

These three possibilities for any real number -- positive, negative, or 0 -- is called the Law of Trichotomy.

Problem 1.   Sketch the graph of  y = x² − 2x −8.  That is, show the x- and y-intercepts.

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

Problem 2.   Sketch the graph of  y = x² + 4x + 4.

Problem 3.   Inspect the graph of  y = x² + 2x − 3 to solve this inequality:

x² + 2x − 3 > 0

The sum and product of the roots

Theorem.   In a quadratic whose leading coefficient is 1:

the sum of the roots is the negative of the coefficient of x;

the product of the roots is the constant term.

That is, if

x² + bx + c = 0,

and the roots are r and s, then

For, if the roots are r and s, then the quadratic is

The coefficient of x is −(r + s), which is the negative of the sum of the roots.  The constant term is rs, which is their product.

Example 5.   Construct the quadratic whose roots are 2 and 3.

 Solution.   The sum of the roots is 5, their product is 6, therefore the quadratic is  x² − 5x + 6.

The sum of the roots is the negative of the coefficient of x.  The product of the roots is the constant term.

Example 6.   Construct the quadratic whose roots are 2 + ,  2 − .