x  5x  4  0
2
( x  1)  25
2
Solving Quadratic
Equations
x
b 
b  4 ac
2
2a
x  100
2
Solving Quadratic Equations
What is the definition
of a solution to a
quadratic equation?
y  ax
c
ax
 bxbx
c  0
2
2
A solution is the
value of x when y = 0.
What are other terms for the solutions to a quadratic equation?
(in some cases)
Graphs and Solutions to Quadratic
Functions: 3 Cases
Two x-intercepts =
two real solutions
(rational or irrational)
ALL QUADRATIC
EQUATIONS CAN
BE SOLVED!
One x-intercept =
one real solution
(always rational)
No x-intercepts =
two complex
solutions
Solving Quadratics
Name the 4 methods of solving quadratics
(x 
)( x 
)0
Method 1: Solving Quadratic Equations by Factoring
ax
ok
2
 bx  c  0
ok
2
1
Let's solve the equation x  7 x  18
need this
to be 0
1. First you need to get it in what we call “standard form"
which means
2
ax  bx  c  0
Factoring is the easiest
2
Subtract 18 x  7 x  18  0
way to solve a
quadratic
equations,
2. Now let's
factor
 x  9  x  2   0 but itthewon’t
left hand
workside
for all
functions,
many= 0
set eachasfactor
x  9  0 or x  2  0 3. Now
beeach
factored!
andcannot
solve for
answer.
x  9 or x   2
Meaning: 2 x-intercepts,
2 real solutions
Method 2: The Square Root: ax2 + c = 0
This method will work for any equation that
doesn’t have a “bx” term, it only has “ax2” or
“a(x-h)2” and a constant. The objective is to get
x2 alone on one side of the equation and then
take the square root of each side to cancel out
the square.
1. Get the "squared stuff" alone
which in this case is the t 2
t  25 25
2
2
t  5
5 t  125 25
2
5
2. Now square
root each side.
Don’t forget that (-5)(-5) = 25 also!
Meaning: 2 x-intercepts, 2 real solutions
5
Let's try another one:
1. Get the "squared stuff" alone
which in this case is the u 2
u 
 49
4
u 
4
4
2. Square root each side.
49 49
22
uu   
4 4
2
2
4
u
4u  
49 409
2
Remember with a fraction you can square root the
top and square root the bottom
DON'T FORGET BOTH THE + AND –
Hey, what about the – under the square root?
7i
Recall i 
numbers!
1 ,
2
Meaning: no x-intercepts, but there are still 2
solutions.
so x equals two imaginary
You try
3 x  36  0
2
Another Example: “a(x-h)2”
1, Get the "squared stuff" alone
(i.e, the parentheses)
 x  1
2

50
 x  1
2
 50  0
2. Now square root each side
and DON'T FORGET BOTH
THE + AND –
25 · 2
x  1  5 2
-1
-1
Meaning: two x-intercepts, but
they are irrational.
Let's simplify the radical
Now solve for x
x  1  5 2
Perfect Square Trinomials: What’s the pattern?
x  bx
2
Add how
much?
c=?
Factored form
2
1
( x  1)
2
4
( x  2)
2
9
( x  3)
2
16
( x  4)
2
b
 
2
2
x  2x 
x  4x 
x  6x 
x  8x 
x  bx 
2
(x 
b
2
2
2
)
2
2
“add half of b squared”
To complete the square and make a perfect square trinomial,_________________
What completes the square?
b  20
b / 2  10
2
x  20 x 
2
x  12 x 
100
___
36
___
2
81/4
x  9 x  ___
b9
 ( x  10)
b   12
b / 2  6
3 x  24 x 
 ( x  6)
2
9 2
 (x  )
2
b/29/2
2
2
No
___
Solution
You can only
complete the
square when a = 1!
Method 4: The Quadratic Formula
The Quadratic Formula is a formula that can
solve any quadratic, but it is best used for
equations that cannot be factored or when
completing the square requires the use of
fractions. It is the most complicated
method of the four methods.
Do you want to see where the formula
comes from?
The Quadratic Formula
ax  bx  c  0
2
This formula comes from completing the square
of a quadratic written in standard form
5. Simplify radical
1. Subtract c and Divide by a
ax  bx   c
2
a
a
x
2a
a
2. Complete the square:
2
2
b
c
b
2
b
b
x  x 
 

2
2
a
a
4a
4a 2
3. Factor left side, combine right side
2
b 

x 
 
2a 

4. Square root each side
b
b  4 ac
2
4a
2
b
2a
2  4 ac
b
b  4 ac
2


24aa
2
6. Get x alone
2
b
b  4 ac
x 

2a
2a
7. Simplify right hand side
x
b 
b  4 ac
2
2a
“x equals opposite b plus or
minus square root of b squared
minus 4ac all over 2a”
x
b 
b  4 ac
2
2a
x
b 
b  4 ac
2
2a
This part of the formula is b 2  4 ac
called the “Discriminant”
The discriminant tells us what kind of solutions we have:
0
One real solution
one x -intercept
(always rational)

Two real solutions
two x-intercepts
(rational or irrational)

two complex
solutions
(no x-intercepts)
The Quadratic Formula
4x  2x  5  0
2
Solve the equation
1. Identify a, b, c
6. Simplify
a=4
b= 2
c=5
2. Plug into the formula
x
 2b 
Notice the
solutions are
complex!
2 
x
 76
=4•19
8
7. Simplify radical
(4)
b(2)  4 ac
22
(5)
x
 2  2 i 19
8
2 a(4)
8. Simplify final answer, if
possible
5. Simplify
x
2 
4  80
8
Meaning: 0 xintercepts, 2
x
complex solutions

212ii 119
9
84
The Quadratic Formula
x  x 1  0
2
Solve the equation
1. Identify a, b, c
6. Simplify
a=
b=
c=
2. Plug into the formula
x
 b
2
b( )2
7. Simplify radical
( )
 4 ac
2 a( )
5. Simplify
( )
8. Simplify final answer, if
possible
The Quadratic Formula
x  x 1  0
2
Solve the equation
1. Identify a, b, c
6. Simplify
a=1
b= 1
c = -1
2. Plug into the formula
x
 1b

b( 1)2 
2
x
7. Simplify radical
( 1 ) (-1 )
4 ac
5. Simplify
1  1  4
2
5
2
2 a( 1 )
x
1 
Already
simplified
8. Simplify final answer, if
possible
Another example
Solve the equation
2x  4x  5  0
2
True or False?
All quadratic equations have solutions.
“No solution” could never be an answer to a
quadratic equation.
TRUE.
You can solve ANY quadratic equation, you just may need to
use a particular method to get to the answers.
True or False?
The solutions (zero, root) to a quadratic
equation are always x-intercepts on its graph.
False
All quadratics have solutions. It is the value of x when y = 0.
But not all quadratics cross the x – axis, so the solutions will not
always be x-intercepts.
8
f x = x2 +4 x+5
g  x = -  x-4  2 -1
6
4
2
-10
-5
Neither of these functions
have x-intercepts, but they
still have two complex
solutions
5
-2
-4
-6
10
15
True or False?
All quadratics equations can be factored.
False
Here are just a few examples of quadratics that cannot be factored:
x 40
2
x  5x  8  0
2
9x  2x 1  0
2
True or False?
The quadratic formula will be provided
to you for the test and final exam.
False. You will need to have the formula memorized.
Which method should you use?
Solve
x  6x  5  0
2
a. (x+1)(x+5)
b. x = -1, x = -5
Stop!
c. x = 1, x = 5
d. no sol
Before you begin to solve this problem, look at the
possible solutions. What method should you use to solve this
problem?
(x 
)( x 
)0
Which method should you use?
Solve 2
x  3x  2  0
a. x 
 3  17
b. x 
3  17
2
2
c. no sol because you cannot factor it
Stop!
Before you begin to solve this problem, look at the
possible solutions. What method should you use to solve this
problem?
(x 
)( x 
)0
Solve
a.
Which method
should you use?
2x  1
2
x
2
b.
x
2
1
2
c. no sol because you cannot factor it
Stop!
Before you begin to solve this problem, look at the
possible solutions. What method should you use to solve this
problem?
(x 
)( x 
)0
Which method
4 x  2 x  1  0 should you use?
Solve
2
2
a. x 
8
b.
x
2i 8
8
8
c.
x
2  2i 2
d.
x
1 i 2
4
8
Stop!
Before you begin to solve this problem, look at the
possible solutions. What method should you use to solve this
problem?
(x 
)( x 
)0
Which method should you use?
9 x  81  0
2
a.
x  3
b. x   9
Stop!
c. x   8 1
Before you begin to solve this problem, look at the
possible solutions. What method should you use to solve this
problem?
(x 
)( x 
)0
Completing the Square: Use #1
This method is used for quadratics that do not
factor, although it can be used to solve any kind
of quadratic function.
x  6x  5  0
2
1. Get the x2 and x term on one
side and the constant term on the
other side of the equation.
2. To “complete the square,”, add
“half of b squared” to each side.
You will make a perfect square
trinomial when you do this.
( x  3) 
2
14
x  3   14
x  6x 9  5 9
2
x  6 x  9  14
2
3. Factor the trinomial
4. Apply the square
root and solve for x
x  3 
14
Completing the Square: Use #2
By completing the square, we can take any
equation in standard form and find its equation
in vertex form: y = a(x-h)2 + k
1. Get the x2 and x term on one
side and the constant term on the
other side of the equation.
2. To “complete the square,”, add
“half of b squared” to each side.
You will make a perfect square
trinomial when you do this.
( x  3) 
2
y  ( x  3)
14
2
x  6x  5  0
2
x  6x 9  5 9
2
x  6 x  9  14
2
3. Factor the trinomial.
4. Write in standard
 14 form.
What is the
vertex?
(  3,  14)