Download Quadratic Equation

Student Academic Learning Services
Page 1 of 7
Quadratic Equations
What is a Quadratic Equation?
A quadratic equation is an equation where the largest power on a variable is two. All quadratic
equations can be written in the form:
Where ‘a’, ‘b’, and ‘c’ can be any real number, including zero. The examples below are all
quadratic equations. Below each equation the values of ‘a’, ‘b’, and ‘c’ are shown.
–
Graphs of Quadratic Equations
When graphing a quadratic equation it is much easier to put it into the following form:
When graphed, a quadratic function will look like a curve in the shape of a „U‟ or an upsidedown „U‟. A few examples of quadratic functions that have been graphed are shown below.
When „y‟ is set to zero, equations like the examples above are formed. Therefore, solutions to
these equations can be found by determining where the function crosses the x-axis.
These graphs suggest that a quadratic equation may have two, one, or zero solutions depending
on how it intersects the x-axis.
Graph of y = x2 + 3x + 2
Has x intercepts at x = -2 and -1
www.durhamcollege.ca/sals
Graph of y = -x2 + 3x + 1
Has no x-intercepts
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/22/2010
Student Academic Learning Services
Page 2 of 7
Factoring Quadratics
There are several different techniques for factoring quadratic equations, and they all apply to
different situations. It is important to be able to factor quadratics fully, as we will see when we
try to solve them without using graphs.
First, for a quadratic with c = 0, or no constant term, such as example 1 below, you can simply
factor the x out normally and you‟re done.
Second, for a quadratic with b = 0, or no “x term”, such as example 2 below, you can use the
difference of squares method. This method will only work out evenly if „a‟ is a perfect square,
and „b‟ is the negative of a perfect square. The middle example has been factored using this
method. To check that it is factored correctly, try expanding the result.
Third, for a quadratic with b ≠ 0, and c ≠ 0, such as example 3 below, you can use a method that
is the reverse of the “FOIL” method for multiplying two binomials. To check that it is
factored correctly, try expanding the result.
Example 1
2
5x + 2x = x(5x+2)
Example 2
Example 3
2
4x – 9 = (2x+3)(2x-3)
x2 + 3x + 2 = (x+2)(x+1)
Difference of Squares
The difference of squares method for factoring quadratics is simple. It makes use of the
following properties:
This property can be used to factor the types of quadratic equations that have b = 0 or the form
of the examples below. This method works best if „a‟ is a perfect square and „c‟ is the negative
of a perfect square but can actually be used on any quadratic as long as b = 0, and c < 0 (i.e. „c‟ is
negative). Below are 4 examples of quadratic equations factored by the use of the difference of
squares method.
x2 – 9 = (x-3)(x+3)
4x2 – 49 = (2x-7)(3x+7)
18x2 – 50 = 2(9x2-25) = 2(3x-5)(3x+5)
x2 – 5 = (x-
www.durhamcollege.ca/sals
)(x+
)
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/22/2010
Student Academic Learning Services
Page 3 of 7
The Reverse of FOIL
For this next method, you must recall the method for multiplying binomials, known as FOIL
(First, Outside, Inside, Last). When you multiply two binomials the usual result is a trinomial,
which is often a quadratic.
When you have a quadratic, it is often possible to use the reverse process to factor it into two
binomials. It is often not easy to do so, but with practice, you get better at it. Have a look at the
examples below, and check to see that the result will expand, by using FOIL, back to the original
quadratic.
x2 + 4x + 3 = (x+3)(x+1)
(x+2)(x+3)
x2 + x – 6 = (x+3)(x-2)
= x2 + 3x + 2x + 6
=(First+Outside+Inside+Last)
= x2 +5x + 6
A Step-by-step Approach
Learning how to “reverse-FOIL” a quadratic isn‟t easy. The best way to think of it is to go back,
and look at the process of FOIL in reverse. Consider the examples above, we have two
binomials that, when multiplied together, yield a quadratic. But what is the best way to get
those two binomials?
Before we delve into solving quadratic equations, there is one other thing you should learn, and
that is the discriminant. Not all quadratics are factorable, and the discriminant is very useful
because it tells us if a quadratic is actually factorable, or if trying to factor it is a waste of time.
The Discriminant
The discriminant is a tool you can use to see for sure whether a quadratic is factorable. With „a,‟
„b,‟ and „c‟ as the previously defined coefficients of the quadratic, the discriminant is calculated
as:


If the discriminant is less than zero, then the quadratic is not factorable.
If the discriminant is greater than, or equal to zero then the quadratic is factorable.
And it‟s just that simple! So use the discriminant whenever you aren‟t sure if a quadratic is
factorable. Otherwise, you may be wasting your time.
www.durhamcollege.ca/sals
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/22/2010
Student Academic Learning Services
Page 4 of 7
Procedure for Factoring
Method 1 - When a = 1:
Use the following expression as an example for method 1:
–
Step 1: Write every possible factorization of the number „c‟.
Factors for (-6)  (6, -1), (-6, 1), (2, -3), (-2, 3)
Step 2: Find a pair of factors that add up to the number „b‟.
Need a pair that adds up to 1
6-1=5
-6 + 1 = -5
2 - 3 = -1
-2 + 3 = 1 BINGO!!
Step 3: The factorization of the quadratic will look like (x+_ )(x+_ ), where the two factors (3
and -2) will be in the blank spaces.
The final solution will then be that
–
Step 4: Check your answer by expanding it out by FOIL.
This method works because, if you FOIL the result, you end up multiplying the two numbers (in
this case 3 and -2) to get „c‟ (-6), and adding them to get „b‟ (1). Note though, that if „c‟ is
negative, then one of the factors will have to be negative, and the other positive. Also, if „c‟ is
positive, then both of the two factors will either be negative, or both of them will be positive.
Method 2 - When ‘a’ is a prime number:
Use the following expression as an example for method 2:
Step 1: Write every possible factorization of the number „c‟.
Factors for 4  (4, 1), (1, 4), (2, 2), (-2, -2)
Step 2: Find a pair of factors of „c‟ that will add up to the number „b‟ AFTER one of the factors
has been multiplied by „a‟.
Need to find a pair that adds up to 7 when one of them is multiplied by 3.
3(4) + 1 = 13
4+ 3(1) = 7 BINGO!!!
Step 3: The factorization of the quadratic will be (ax+_)(x+_), where the two factors will be in
the blank spaces.
The final solution will then be that
www.durhamcollege.ca/sals
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/22/2010
Student Academic Learning Services
Page 5 of 7
Step 4: Check your answer by expanding it out by FOIL.
The only difference here is in Step 2, where now you have to worry about the number in front
on the x-squared. Because that number is prime, you know that the factorization will look
something like this: (ax + _)(x + _), so that the “First” operation in FOIL results in the right
coefficient for x-squared.
As a result, the „a‟ value gets multiplied into one of the two factors of „c‟ before the factors are
added together.
Method 3 - When ‘a’ is not a prime number:
Use the following equation as an example of method 3:
Step 1: Check to see if you can factor a whole number from the entire equation to make it a
little bit easier. If it does factor, forget about the common factor until the end of the question,
and just worry about the quadratic inside the brackets.
16x2 + 54x +18  2 (8x2 + 27x +9)
Step 2: Write every possible factorization for the number „a‟ and „c‟.
Factors for 8  (8,1), (1,8), (2,4) (4,2)
Factors for 9  (9,1), (1,9), (3,3)
Step 3: Find the factors of „c‟ that when multiplied by some factors of „a‟ will add up to „b‟
Need to find the pair of factors of 9 that, when multiplied by a pair of factors of 8, will add up
to 27.
(8)9 + (1)1 = 73
(8)1 + (1)9 = 17
(8)3 + (1)3 = 27 BINGO!!
Step 4: The factorization of the quadratic will look similar to that of (ax + _ )(bx +_ ), where
the factors of „c‟ will be the blank spots and „a‟ and „b‟ will be the factors of „a‟ from the
question. If the question had a common factor from the start, it is time to bring that back as
part of the question.
The final solution will then be that
Step 5: Check your answer by expanding with FOIL.
As you can probably see, this method involves the most work, because of all the extra factor
pairs you must check in step 3. In this case, we were lucky, and the third try produced the
desired result, where as there were 9 other combinations we could have tried.
www.durhamcollege.ca/sals
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/22/2010
Student Academic Learning Services
Page 6 of 7
Solving Quadratic Equations
To solve a quadratic equation of the form ax2 + bx + c = 0 all you have to do is factor it, and
then solve using each individual factor. This usually results in two different solutions.
In the next example we have one of our previously factored quadratics set equal to zero.
By factoring the quadratic like this, we can now see two solutions to the equation. After all, if
either of (x - 2) or (x + 3) is equal to zero then the whole left side will be equal to zero. So, we
can find two different solutions to this equation by solving the two linear equations
x-2=
0 and x + 3 = 0.
We end up with the solutions of x = 2, and x = -3, and that is what we were looking for!
–
–
–
–
No factors!
Check the discriminant!
The Quadratic Formula
The quadratic formula is a brute-force formula for solving any quadratic. When there is no
factorization to be found for a quadratic, the quadratic formula is a very useful tool. It is
extremely useful if the values of „a,‟ „b,‟ and „c‟ are very large giving them many more factors.
The quadratic formula is fairly straight-forward. The one thing that you have to remember is
that, because of the „+/-‟ sign, you will usually end up with two solutions: one for the „+‟ and
one for the „-‟.
The Formula is shown at the top of the next page.
Look at the part in the square root. It should look very familiar. That is the discriminant that we
discussed above! It can now be seen that if the discriminant is a negative number, we will have
to take the square root of a negative number. That is why no solutions exist if the discriminant
is less than 0.
www.durhamcollege.ca/sals
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/22/2010
Student Academic Learning Services
Page 7 of 7
Examples of Using Quadratic Formula
It is best to look at examples to see how this formula works. There is a lot of work involved
when using the quadratic formula, but the good part is that there will always be a solution (as
long as the discriminant is greater than 0) no matter how ugly it looks.
x2 + 5x + 4 = 0
3x2 – 5x – 7 = 0
In this case, a=1, b=5, and c=4
In this case, a=3, b=-5, and c=-7
x
x
x
x
x
x
x
b
b2
2a
5
52 4(1)( 4)
2(1)
5
25 16
2
9
5
4ac
x
x
x
x
2
5 3
,x
2
2
8
,x
2
2
1, x
4
5 3
2
x
x
x
b2
2a
b
5
4ac
( 5) 2 4(3)( 7)
2(3)
5
25 84
6
5
109
6
5 10.44
5 10.44
,x
6
6
15.44
5.44
,x
6
6
2.57, x 0.91
In this step, we split into two separate solutions,
one using the plus, and one using the minus.
www.durhamcollege.ca/sals
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/22/2010