1. Chapter 7 Quadratic Equations and Functions
Algebra 2
Chapter 7 Quadratic Equations and Functions

2. 7-7 Writing Quadratic Equations and Functions
WARMUP:
Solve the following:

3. 7-7 Writing Quadratic Equations and Functions
GOAL: To learn the relationship between the roots and coefficients of a quadratic equation. To write a quadratic equation or function using information about the roots or the graph.

4. 7-7 Writing Quadratic Equations and Functions
Our goal, in a nutshell, is to be able to write a quadratic equation if given two roots.
Let’s solve this quadratic equation:
2x2 + 2x – 12 = 0
It is factorable:
2(x2 + x – 6) = 0

5. 7-7 Writing Quadratic Equations and Functions
2(x2 + x – 6) = 0
It is still factorable:
2(x + 3)(x – 2) = 0
So:
x = -3 or x = 2.

6. 7-7 Writing Quadratic Equations and Functions
So let’s work backwards. Say we are given that the roots to some quadratic equation are:
x = -3 or x = 2
We can easily reason that our equation is:
(x + 3)(x – 2) = 0
And then in standard quadratic form:
x2 + x – 6 = 0
There may have been a coefficient factored out, so: a(x2 + x – 6) = 0, a≠0

7. 7-7 Writing Quadratic Equations and Functions
By that thinking, we can generalize this:
If r1 and r2 are the roots of a quadratic equation, then:
(x – r1)(x – r2) = 0
If we foil this, we get:
x2 – (r1 + r2)x + r1r2 = 0
And this is a theorem:

8. 7-7 Writing Quadratic Equations and Functions
Theorem:
A quadratic equation with roots r1 and r2 is:
x2 – (r1 + r2)x + r1r2 = 0
or a[x2 – (r1 + r2)x + r1r2]= 0

9. 7-7 Writing Quadratic Equations and Functions
Example:
Find a quadratic equation with roots 5 and -1:
r1 = 5 and r2 = -1
x2 – (r1 + r2)x + r1r2 = 0 so
x2 – (5 + -1)x + (5)(-1) = 0
x2 – 4x – 5 = 0

10. 7-7 Writing Quadratic Equations and Functions
Another example. Let’s look at example 1 in the book, on page 339:

11. 7-7 Writing Quadratic Equations and Functions
Theorem:
If r1 and r2 are the roots of a quadratic equation ax2 + bx + c = 0, then
r1 + r2 = sum of roots = and
r1r2 = product of roots =
This theorem is useful when checking solutions.

12. 7-7 Writing Quadratic Equations and Functions
More examples.

13. 7-7 Writing Quadratic Equations and Functions
HOMEWORK!