This document discusses solving quadratic equations using the quadratic formula. It begins by reviewing quadratic equations and explaining that while some can be solved by factoring, the formula can solve all quadratic equations. It then introduces the quadratic formula, explaining that it requires identifying the a, b, and c values from the quadratic equation in standard form. The document provides several examples of setting up and solving quadratic equations using the formula. It emphasizes that the formula always provides two solutions and concludes by noting another step may be needed to simplify solutions in their simplest form.

1. In this last part of the quadratic lesson, we will solve quadratic equations by
using the Quadratic Formula. The formula is on the formula page so you don’t
have to memorize it—just use it. There will be two answers.
Using the Quadratic Formula for solve quadratic equations
First, let’s review just what we mean by a quadratic equation. Here are two.
Notice that the highest exponent is 2 rather than 1 as in 2x + 1 = 4 (linear equation).
Remember that some quadratic equations can be solved by factoring which we
discussed in the last session (Part 2). Not all quadratics will factor so the formula is the only way.
However, all quadratic equations can be solved by using the formula. It solves them all.
The unfortunate part is that the formula can be a little hard to use. So, here goes.
The first thing to know about using the formula is that the equation must be in
standard form. That means that one side must be 0 and the terms on the other
side must be in descending order of the powers of x.
This is in standard form. Notice that the x2 is first, the x term
is second and the number (the constant) is third.
This is not in standard form yet. We must subtract 1 from both sides
and rewrite it as 4x2 -3x -1 = 0 I have seen this type on a practice test.

2. To begin with, let’s look at the formula. There are three values we need: a, b, and c.
Here is the equation in standard form. Notice that a is the number in from of x2, b is
the number with the sign in front of x and c is the number alone.
To solve the equation, we just plug in the a, b, and c numbers in the formula and
simplify.
First, we have to practice identifying a, b, and c.
a = 1, b = 2, c = -8
a = 1, b = -5, c = 2
a = 1, b = 3, c = -7
a = 2, b = -5, c = 2

3. Let’s try another one. This one is in standard form already.
a = 1, b = -3, and c = - 4 I figure out the b2 part out to the side.
Also, the 9 – (-16) part on scratch paper. That is where the 25 comes from.
Also, I think about the –b (the opposite of b) part mentally so there is not so much
to write down.

4. Now we will solve some problems using the formula. For some worked out examples,
see page 156 in your book.
The two answers are 5 and 2.

5. Let’s try another one.
a = 3, b = 5, c = 1
The answers may look strange, but this is the best we can do for this one.

6. Now let’s try one that starts out not in standard form.
Rewrite it as 4x2 -3x -1 = 0 so that a = 4 b = -3 and c = -1
So, the two answers are 1 and -1/4.

7. Solving quadratic equations is a very hard task the first time students look at
it. Please keep trying by doing some of the problems over and doing some
from the book page 157 (the puzzle page). There is one more complication
that I did see on one of the practice tests I tried. It has to do with simplifying
the answer. See below. We need to do the division problems (still not
explained) for this to make complete sense. Just know this is the equivalent
to simplifying fractions to lowest terms.
Divide all terms in the top and bottom
by 4
Divide all terms top and bottom by
4.