Quadratic Equations

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Quadratic Equations
Three simple methods for solving quadratic
equations
By Matt D. and Brian V.
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THE THREE METHODS
Factoring (foil)
Completing the Square
Quadratic equation
(Click one to begin)
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FACTORING
EXAMPLE EQUATION
2x2 10x 8-4
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STEP 1
SET THE EQAUTION EQUAL TO ZERO
2x210x8-4
To set the equation equal to zero simply add four
to both sides of the equation.
Click for answer
2x2 10x 12 0
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STEP 2
TAKE OUT THE GREATEST COMMON FACTOR (GCF)
2x2 10x 120
To complete the next step of this method you take
out the Greatest Common Factor (GCF) or largest
number that is a factor of every term in the
equation. For this equation the GCF is 2, because
it is the largest factor of ALL the terms.
Click for answer
2(x2 5x 6) 0
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STEP 3
SET UP TWO SETS OF PARENTHESIS ( ) ( )
( ) ( )
Easy enough
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FOIL
Step 3 continued
The next thing that you need to do to solve the
equation is learn how to FOIL. To set up your
FOIL equation you place the letters of the word
FOIL to the corresponding places shown below.
( ) ( )
F
L
F
L
O I
I O
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STEP 4
CONTINUE FOILING
The next step in FOILing the equation is to
take the first term in the trinomial and take
half of the exponent and put the variable and
exponent in the F places in the equation. The
trinomial in this equation is (x2 5x 6), so
the first term would be x2. Half of the exponent
(2) would be 1, so you would put x1 (x) in the
F places.
Click for answer
F L
F L
( x ) (x )
O I
O I
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STEP 5
WHAT OPERATION TO USE?
After you have the first term in each of your
Parenthesis you need to put an operation, either
addition or subtraction, in the equation. You
need to know whether or not the operations will
be the same, and what they will be, addition or
subtraction. To do this you look at the second
sign, or operation, in the trinomial. If it is
positive (addition) the signs will be the same
(whatever the first sign is), if it is negative
(subtraction) the signs are different. In our
example problem the trinomial is (x2 5x 6),
so the second sign is addition. We now know that
the signs will be the same. The first sign is
positive, so we know that both sets of
parenthesis will have addition for a sign.
Click for answer
(x ) (x )
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STEP 6
FIND THE SECOND TERM
Now that you have the first term, and operation
of both sets of parenthesis, you need to find the
second term for each set. To do this you have to
list the factors of the last term in the
trinomial. In our equation the last term in the
trinomial is 6. (x2 5x 6).
Click for answer
6 1 6 2 3
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STEP 7
WHAT TO DO WITH THE FACTORS?
If the second sign in your trinomial is addition,
you find the pair of factors that add to get the
second term in the trinomial. If the second sign
is subtraction, you find the pair of factors that
subtract to get the second term. The second term
in our problem is addition, so you would have to
find the factor pair of six, that adds to get
five. If there are no factor pairs that add, or
subtract to equal B, the equation is PRIME.
(x2 5x 6)
Click for answer

• 6
• 6 7
• 2 3 5

3 and 2 are the factor pair you want, because
they add to equal 5.
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STEP 7
Leading coefficient other than one
(Continued)
If the leading coefficient of an equation is a
value other than one, you have to find the
factors of (a) (first coefficient), and (c)
(third coefficient). You multiply the factors of
(a) by the factors of (c), and find a combination
that either add or subtract to equal (b)
(depending on the sign). To determine where the
factors a placed in the two sets of parenthesis
you find the greater product of the two sets. You
take the last value of the pair of factors with
the greatest product, and place it in the last
spot in the parenthesis that has a corresponding
sign with that in the first spot in the equation.
Click for answer
Factors of 4 Factors of 2 (1)
(2) 2 (4) (1) 4 2 and 4
add to equal 6

• 4
• 1 4
• 2

2 1 2
(4y2)(y1)
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STEP 8
FINISHING UP
This factor pair is going to be the last thing
you put into your set of parenthesis to solve the
problem. Once you have found your factor pair you
have to decide which value goes into which
parenthesis. To do this you have to look at the
signs that you put in your parenthesis in step 5.
If the signs are the same, place the factor pairs
in the L spots. If the signs are different,
place the greatest factor of the pair in the
parenthesis that contains the same sign as the
first of the trinomial. If the first sign of the
trinomial is () and your factor pair is (6,3)
then you place the 6 in the parenthesis with the
() sign. In our problem the signs are both
addition, so you place one value in each
parenthesis.
Click for answer
2(x 3)(x 2)
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STEP 9
Finding xs values
Now that you have your new equation 2(x3)(x2)
you need to find out what x equals. To do this
you set the equation equal to zero, and x will
then equal the opposite of whatever is being
added to it, or subtracted from it, in each
parenthesis.
2 (x 3) (x 2) 0
Click for answer
x -3, -2 This is your final answer
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CHECKING YOUR ANSWER
To check you answer you go use the FOIL set up,
and multiply the value under each letter in the
first parenthesis, by the corresponding value of
the same letter in the second parenthesis.
F L F
L
(x 3) (x 2)
O I I
O
F (x)(x) x2 O (x)(2) 2x I (3)(x) 3x L
(3)(2) 6
Once you have these values, you add them all
together. If the equation you get corresponds to
the original equation your answer is correct.
Click for answer
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COMPLETING THE SQUARE
ax2 bx c 0
EXAMPLE EQUATION
x2 7x 10 0
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STEP 1
MOVE THE CONSTANT
x2 7x 10 0
The first step in solving an equation using this
method is to move the constant to the right side
of the equal sign. The terms with variables stay
on the left side of the equal sign. In this
equation, you move the constant to the right side
of the equal sign by adding it (10) to both sides.
Click for answer
x2 7x 10
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STEP 2
LEADING COEFFICIENT
In some equations, the leading coefficient will
be a value other than one. In the equation 2x2
4x8, the leading coefficient is 2. If you have
an equation like this, you divide every term by
the leading coefficient.
2x2 4x 8
(Divide every term by 2)
Click for answer
x2 2x 4
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STEP 3
USING B
x2 7x 10
The next step in solving this equation involves
using the formula for completing the square,
which is, ax2 bx c 0. The next thing that
you would do is to take (1/2 b)2 and add it to
both sides of the equation. In our equation b is
7. Half of 7 is (7/2). (7/2)2 is (49/4)
Click for answer
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STEP 4
SIMPLIFY
The next step you take is to simplify the right
side of the equation. To do this in this
equation, you would add 10 to the quantity of
(49/4). In order to add, the two values must have
the same denominator. You could change 10 to
(40/4). This would give you common denominators
so you could then simply add the terms.
Click for answer
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STEP 5
REWRITE
Once you have simplified the right side of the
equation, the next thing to do is rewrite the
left side of the equation as a binomial squared.
To do this, you take the quantity of x plus or
minus (depending on what the first sign is) half
of b and square that quantity. The first sign in
this equation is subtraction, so we would
subtract half of b from x and square the quantity.
Click for answer
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STEP 6
SQUARE ROOT
After you rewrite the left side of the equation,
and the equation looks similar to the one shown
above, the next thing to do is take the square
root of both sides of the equation. The left side
of the equation is a quantity squared, so the
square root of it is simply that quantity. For
the right side of the equation, you take the
square root of the numerator and the denominator
and keep it as a fraction. Since the square root
of a term is a positive and negative value, the
left side will equal or - the right.
Click for answer
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STEP 7
SOLVE
The final step in solving the equation using
completing the square is to solve for x. To do
this, the first thing you need to do, is get x
alone on one side of the equation. Then you would
move the other values on the same side as x to
the other side of the equation. In this problem
you would ADD (7/2) to both sides of the
equation. You would then have x alone and could
simplify to get your final answer. In some
equations the number inside the square root sign
will be negative. If your equation has an answer
like that, you could leave it as it is, or write
no rational roots. You could write this because
the square root of a negative number is not
rational.
Click for answer
(SIMPLIFY)
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QUADRATIC EQUATION
EXAMPLE EQUATION
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STEP 1
Learning the formula
Using the Quadratic equation (shown above) you
can set up your equation to work in the quadratic
formula (shown below).The example equation is set
up to work in the quadratic formula.
Click for answer
x
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STEP 2
Setting up the formula
Now that the equation is set up in the form of
the correct form you can substitute the values of
your equation for the corresponding variables in
the quadratic formula (shown below)
Click for answer
a4 c2 b(-3)

x
x
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STEP 3
Simplify
x
After you have put all the values in the equation
into the quadratic formula the next step is to
simplify (using order of operations).
Click for answer
x


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STEP 4
Finishing up
x
In this equation the final answer contains the
square root of a negative number. The answer to
this would be not be rational, so therefore your
final answer could be no rational roots or
leave it as it is. This is only the case in
problems where you have a square root of a
negative number. If your answer contains the
square root of a positive number, you could
simplify (if it contains a perfect square) or
leave it as is.
x
Can be reduced to
(Click for answer)
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THE END
Practice problems
Menu
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PRACTICE PROBLEMS
Click for problem 1
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1
Solve by factoring
ANSWERS (choose one)
A
C
B
D
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WRONG ANSWER
33
CORRECT!
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2
Solve by using the quadratic formula
A
C
B
D
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WRONG ANSWER
36
CORRECT!
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3
Solve by completing the square
A
C
B
D
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WRONG ANSWER
39
CORRECT!
40
4
Solve by factoring
A
B
C
D
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WRONG ANSWER
42
CORRECT!
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5
Solve by using the quadratic formula
Answer
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Answer
45
6
Solve by completing the square
Answer
46
Answer
47
7
Solve by factoring
C
A
B
D
48
WRONG ANSWER
49
CORRECT!
50
8
Solve by using quadratic formula
Answer
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ANSWER
(no rational roots)
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9
Solve by completing the square
Answer
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ANSWER
(no rational roots)
54
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A
C
B
D
Prime
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WRONG ANSWER
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CORRECT!
The answer to this equation is PRIME (b), because
no factors of 1, add to equal 3.