Algebra 2

Chapter 7 Quadratic Equations and Functions

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

2

2

2

5 4 0

3 8 5 0

3( 5) 0

x x

w w

x

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.

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

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.

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 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:

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

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

7-7 Writing Quadratic Equations and Functions Another example. Let’s look at example 1 in

the book, on page 339:

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.

b

a

c

a

7-7 Writing Quadratic Equations and Functions More examples.

7-7 Writing Quadratic Equations and Functions HOMEWORK!