Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Methods for Solving Quadratic Equations
Quadratics equations are of the form ax 2 + bx + c = 0, where a ≠ 0
Quadratics may have two, one, or zero real solutions
...
FACTORING
Set the equation equal to zero
...

Example:
x 2 = −5 x − 6
Move all terms to one side
Factor
Set each factor to zero and solve

x2 + 5x + 6 = 0
( x + 3)( x + 2) = 0
x+3= 0
x+2=0
x = −3
x = −2

2
...
Then take the square root of both sides
...
)
Example 1:

x 2 − 16 = 0

Move the constant to the right side

x 2 = 16

Take the square root of both sides

x 2 = ± 16
x = ±4, which means x = 4 and x = −4

Example 2:

2( x + 3) 2 − 14 = 0

Move the constant to the other side

2( x + 3) 2 = 14

Isolate the square

( x + 3) 2 = 7

Take the square root of both sides

(divide both sides by 2)

( x + 3) 2 = ± 7
x+3= ± 7

Solve for x

x = −3 ± 7
This represents the exact answer
...


3
...


x 2 + 6 x − 11 = 0
Example:
**Important: If a ≠ 1 , divide all terms by “a” before proceeding to the next steps
...

Decimal approximations can be found using a
calculator
...
QUADRATIC FORMULA
Any quadratic equation of the form ax 2 + bx + c = 0, where a ≠ 0 can be solved for both real and
imaginary solutions using the quadratic formula:

x=

− b ± b 2 − 4ac
2a

x 2 + 6 x − 11 = 0

Example:

( a = 1,

b = 6,

c = −11)

Substitute values into the quadratic formula:

− 6 ± 62 − 4(1)(−11)
x=
2(1)

x=

−6±4 5
2

→

→

x=

− 6 ± 36 + 44
2

→

x=

− 6 ± 80
2

simplify the radical

x = −3 ± 2 5 This is the final simplified EXACT answer



---