Solving Quadratic
Equations by the
Quadratic Formula
Do Now:

Solve for x by factoring first:
x  2x  8  0
2
THE QUADRATIC FORMULA
1. When you solve using completing the square
on the general formula ax 2  bx  c  0
you get:
2
x 
b 
b  4 ac
2a
2. This is the quadratic formula! Equation must
be in standard form first-ax2+bx+c=0
3. Just identify a, b, and c then substitute into
the formula.
Do Using the Formula:

List a, b, c first:
x  2x  8  0
2
Do Using the Formula:

List a, b, c first:
x  2x  8  0
2
x
2
36
2
x
26
2
 4, x 
26
2
 2
{4, 2}
Graph the following:

Trace to the approximate roots:
x  4 x  10  0
2
Do Using the Formula:

List a, b, c first:
x  4 x  10  0
2
Do Using the Formula:

List a, b, c first:
x  4 x  10  0
2
x
4
56
2
x
4
56
2
 4, x 
2  2 14
1
2
14
Practice:

Solve:
2x  4x  4  0
2
Practice:

Solve:
2x  4x  4  0
2
x
4
(4 )  4 (2 )(4 )
2
2 (2 )
x
4
48
4

44
4
3
 1
3
WHY USE THE
QUADRATIC FORMULA?
The quadratic formula allows you to solve
ANY quadratic equation, even if you
cannot factor it.
An important piece of the quadratic formula
is what’s under the radical:
b2 – 4ac
This piece is called the discriminant.
WHY IS THE DISCRIMINANT
IMPORTANT?
The discriminant tells you the number and types of answers
(roots) you will get. The discriminant can be +, –, or 0
which actually tells you a lot! Since the discriminant is
under a radical, think about what it means if you have a
positive or negative number or 0 under the radical.
WHAT THE DISCRIMINANT
TELLS YOU!
Value of the Discriminant
Nature of the Solutions
Negative
2 imaginary solutions
Zero
1 Real Solution
Positive – perfect square
2 Reals- Rational
Positive – non-perfect
square
2 Reals- Irrational
Example
Find the value of the discriminant and describe the nature of the
roots (real,imaginary, rational, irrational) of each quadratic
equation. Then solve the equation using the quadratic formula)
1.
2 x  7 x  11  0
2
Discriminant =
a=2, b=7, c=-11
b  4ac
2
(7 )  4 (2 )(  1 1)
2
49  88
Discriminant = 1 3 7
Value of discriminant=137
Positive-NON perfect
square
Nature of the Roots –
2 Reals - Irrational
Students will review for the quadratics test.
The discriminant-b2-4ac

X2 -4x + 4 = 0

Disc =

X2 -2x + 10 = 0

Disc=

X2 +6x + 8 = 0

Disc =

2X2 -5x + 1 = 0

Disc =
The discriminant

X2 -4x + 4 = 0

Disc = 0

X2 -2x + 10 = 0

Disc= -36

X2 +6x + 8 = 0

Disc = 4

2X2 -5x + 1 = 0

Disc = 17
What is the nature of these roots?
The discriminant

X2 -4x + 4 = 0


X2 -2x + 10 = 0


X2 +6x + 8 = 0

2
 2X
-5x + 1 = 0

0, 1 real root
Negative = (2
imaginary roots)
pos perfect square=
(2 rational roots)
pos not perfect=2
irrational roots
Do Now

Solve for x:
x  6x  5  0
2

Solve for x:
2x  7x  3  0
2
Now add and multiply the 2
roots:

Solve for x:
x  6x  5  0
2

Solve for x:
2x  7x  3  0
2
5,1
5 1 6
(5 )(1)  5
 3, 
1
2
2x  7x  3  0
2
 3, 
1
2
Sum and product of the roots:

Sum of the roots:
3  
1
2

7
2
Product of the roots:
(3)(
1
2
)
3
2
Sum and product of the roots:

Sum of the roots
Product of the roots:
b
c
a
a
x  6x  8  0
2
Example:
Sum = 6
b
6 c
8
 , 
Product = 8
using sum and product:
Do now:

Hint: we know what
the sum is:



Given:
X2 -2x + c = 0
and one root is
5 find the other
root
Find c
using sum and product:
Do now:

Hint: we know what
the sum is:


Given:
X2 -2x + c = 0
and one root is
1/3 find the
other root
using sum and product:
Do now:

Hint: we know what
the product is:


Given:
X2 -bx + 10 = 0
and one root is
2, find the
other root
Use the discriminant

For what value of c
does the equation
have equal roots:
x  4x  c  0
2
Use the discriminant

For what value of c
does the equation
have equal roots:
x  4x  c  0
2
b  4 ac  0
2
(4 )  4 (1)(c )  0
2
16  4 c  0
16  4 c
c4
Use the discriminant

For what value of c
does the equation
have imaginary
roots:
x  4x  c  0
2
Use the discriminant

For what value of c
does the equation
have imaginary
roots:
(less than zero)
x  4x  c  0

2
b  4 ac  0
2
(4 )  4 (1)(c )  0
2
16  4 c  0
4 c  16
c4
Quadratic equation knowing
sum and product
X2 – sum (x) +product =
0

Write the equation
when the sum is - 6
and the product is 8
x  6x  8  0
2
Opposite of the sum, same as the product
using sum and product:
to write an equation



Given the roots:
2, -5
Write the equation:

Hint: we can find
the sum and the
product.
using sum and product:
to write an equation



Given the roots:
2, -5
Write the equation:


Sum= -3
Product = -10
x  sum ( x )  product  0
2
x  3 x  10  0
2
using sum and product:
Do now:



Given the roots:
2 + 6i,2-6i
Write the equation:

Hint: we can find
the sum and the
product.
Quadratic equation knowing
sum and product
X2 – sum (x) +product = 0

Write the equation
when the sum is 4
and the product is
40.
x  4 x  40  0
2
Quadratic equation knowing
sum and product
X2 – sum (x) +product = 0

Write the equation
when the sum is 2/3
and the product is 4
Quadratic equation knowing
sum and product
X2 – sum (x) +product = 0
x 
2
2
x4 0
3
3x  2 x  12  0
2


Write the equation
when the sum is 2/3
and the product is 4
Clear the fraction!
using sum and product:
Do now:

Given the roots:
2
3
2

Write the equation:

Hint: we can find
the sum and the
product.
using sum and product:
Do now:

Given the roots:
2
3
2

Write the equation:

the sum is 2 and the
product is 1/4.
Quadratic equation knowing
sum and product
X2 – sum (x) +product = 0


Write the equation
when the sum is 3/5
and the product 2/3
Clear the fraction!
Quadratic equation knowing
sum and product
X2 – sum (x) +product = 0
x 
2
3
5
x
2
3


0
Write the equation
when the sum is 3/5
and the product -2/3
Clear the fraction!
Quadratic equation knowing
sum and product
X2 – sum (x) +product = 0
x 
2
3
5
x
2
0
3
15 x  9 x  10  0
2


Write the equation
when the sum is 3/5
and the product -2/3
Clear the fraction!
Example #1- continued
Solve using the Quadratic Formula
x  6 x  28  0
2
a  1, b  6, c  8, b  4 ac  (6 )  4(1)(8)  4
2
b 
b  4 ac
2
2a
6
( 6 )  4(1)(8)
2
2(1)
6
4
2 R eals - rational
2
62
 4,
62
2
Example 2
Find the value of the discriminant and describe the nature of the
roots (real,imaginary, rational, irrational) of each quadratic
equation. Then solve the equation using the quadratic formula)
1.
2 x  7 x  11  0
2
Discriminant =
a=2, b=7, c=-11
b  4ac
2
(7 )  4 (2 )(  1 1)
2
49  88
Discriminant = 1 3 7
Value of discriminant=137
Positive-NON perfect
square
Nature of the Roots –
2 Reals - Irrational
Example #2- continued
Solve using the Quadratic Formula
2 x  7 x  11  0
2
a  2, b  7, c   11
b 
b  4 ac
2
2a
7 
7  4(2)(  11)
2
2( 2)
7 
137
4
2 R eals - Irrational
Solving Quadratic Equations
by the Quadratic Formula
Try the following examples. Do your work on your paper and then check
your answers. Find the sum and product of the roots:

1. x  2 x  63  0
2
2. x  8 x  84  0
2
3. x  5 x  24  0
2
Sum
Product
Solving Quadratic Equations
by the Quadratic Formula
Try the following examples. Do your work on your paper and then check
your answers. Find the sum and product of the roots:

1. x  2 x  63  0

2. x  8 x  84  0

3. x  5 x  24  0

2
2
2
Sum
1. -2
2. -8
3. 5
Product
-63
-84
-24
Solving Quadratic Equations
by the Quadratic Formula
Try the following examples. Do your work on your paper and then check
your answers. Find roots:

1. x  2 x  63  0
2
2. x  8 x  84  0
2
3. x  5 x  24  0
2
Roots:
Solving Quadratic Equations
by the Quadratic Formula
Try the following examples. Do your work on your paper and then check
your answers.


1. x  2 x  63  0
1. 9,7
2. x  8 x  84  0
2 .(6 ,  1 4 )
2
2

3 .  3, 8
3. x  5 x  24  0
2

Review-Students will review for the quiz on quadratic equations
Do Now:

Find the nature of
the roots for the
following quadratic
equations:
1. x  2 x  6  0
2
2. x  8 x  2  0
2
3. x  4 x  5  0
2

Hint: use the
discriminant.
Solving Quadratic Equations
by the Quadratic Formula
Try the following examples.

1. x  2 x  3  0
2
2. x  8 x  12  0
2
3. x  4 x  13  0
2
Roots:
Solving Quadratic Equations
using complete the
Try the following example.

Solve using complete the
square:
1. x  2 x  10  0
2