Transcript Solving Quadratic Equations by Finding Square Roots

Solving Quadratic Equations
by Finding Square Roots
This lesson comes from
chapter 9.1 from your
textbook, page 503
What is a square root?
If a number square (b2) = another
number (a), then b is the square root
of a.
Example: If 32 = 9, then 3 is the
square root of 9
Some basics…
All positive numbers have two
square roots
The 1st is a positive square root,
or principal square root.
The 2nd is a negative square root
Square roots are written with a
radical symbol
You can show both square roots
by using the “plus-minus” symbol
±
Find the square root of numbers
64  8
 64  8
0 0
 0.25  .5
4
(undefined )
Perfect Squares: Numbers whose square roots
are integers or quotients of integers.
1 1
4 2
9 3
16  4
25  5
36  6
49  7
64  8
81  9
100  10
121  11
144  12
169  13
Evaluate a Radical Expression
Evaluate b 2  4ac when a  1, b  2, and c  3.
b 2  4ac  (2) 2  4(1)( 3)  4  4(3)
 4  12  16  4
Quadratic Equations
Standard form: ax2 + bx + c = 0
a is the leading coefficient and cannot
be equal to zero.
If the value of b were equal to zero,
the equation becomes ax2 + c = 0.
We can solve equations is this form by
taking the square root of both sides.
Key Concepts
When x2 = d
If d > 0, then x2 = d has two solutions
If d = 0, then x2 = d has one solution
If d < 0, then x2 = d has no real
solution
Solving quadratics
Solve each equation.
a. x2=4 b. x2=5 c. x2=0
d. x2=-1
x2=4 has two solutions, x = 2, x = -2
x2=5 has two solutions, x =√5, x =- √5
x2=0 has one solution, x = 0
x2=-1 has no real solution
Solve by rewriting equation
Solve 3x2 – 48 = 0
3x2 – 48 + 48 = 0 + 48
3x2 = 48
3x2 / 3 = 48 / 3
x2 = 16
After taking square root of both sides,
x=±4
Equation of a falling object
When an object is dropped, the speed
with which it falls continues to increase.
Ignoring air resistance, its height h can be
approximated by the falling object model.
h  16t  s
2
h is the height in feet above the ground
t is the number of seconds the object has been falling
s is the initial height from which the object was dropped
Application
An engineering student is in an “egg
dropping contest.” The goal is to
create a container for an egg so it can
be dropped from a height of 32 feet
without breaking the egg. To the
nearest tenth of a second, about how
long will it take for the egg’s container
to hit the ground? Assume there is no
air resistance.
h  16t  s
2
The question asks to find the time it
takes for the container to hit the
ground.
Initial height (s) = 32 feet
Height when its ground (h) = 0 feet
Time it takes to hit ground (t) =
unknown
h  16t  s
2
Substitute
0 = -16t2 + 32
-32 + 0 = -16t2 + 32 – 32
-32 = -16t2
-32 / -16 = -16t2 / -16
2 = t2
t = √2 seconds or approx. 1.4 seconds