Week 24: Quadratics by factoring, and value problems

Field Book

86 to 89

Due

Test 21 due

Method

Paper first, every step

This is the week the year turns. Everything before it solved equations where x appeared once. A quadratic has x squared, which means it usually has two answers, and the trick for finding them is beautifully simple: if two things multiply to zero, one of them has to be zero. That single sentence is the whole method.

What you have to be able to do

Solve a quadratic equation by factoring
Test whether a relation is a function
Set up a value problem

The procedures

Solving a quadratic by factoring

Get everything on one side so the other side is zero. This step is not optional and it is where most errors start.
Factor the left side into two binomials.
Set each factor equal to zero separately and solve. You get two answers, and both are real answers.

The vertical line test

Draw or imagine a vertical line sweeping across the graph.
If it ever touches the graph in two places at once, the relation is not a function.
One input, one output. That is the whole requirement.

Value problems

Two columns again: how many, and what each is worth.
Number times value equals total value, one row per kind of item.
Two equations, two unknowns, then solve the system.

Worked

WORKED

Solve x² + 5x + 6 = 0.
The right side is already zero. Factor the left: two numbers multiplying to 6 and adding to 5 are 2 and 3.
(x + 2)(x + 3) = 0, so either x + 2 = 0 or x + 3 = 0.
ANSWER: x = −2 or x = −3

WORKED

Solve x² − 9 = 0.
Difference of two squares: (x + 3)(x − 3) = 0.
Set each factor to zero.
ANSWER: x = 3 or x = −3

Terms

Term	Meaning
quadratic equation	an equation whose highest power of the variable is 2
zero product property	if a product is zero, at least one factor is zero
root	a solution of an equation, a value that makes it true

Check yourself

1. Solve x² + 7x + 10 = 0

2. Solve x² − 25 = 0

3. Why must one side be zero before you factor?

reveal answers

x = −2 or −5 · x = 5 or −5 · because the zero product property only works against zero

This page is the summary, not the lesson. The full working, every step of every example, is on the board in class. Come here to remember what the week was about. Go there to learn how it is done.