11.1 Solving Quadratic Equations by the Square Root Property • Square Root Property of Equations: If k is a positive number and if.
Download
Report
Transcript 11.1 Solving Quadratic Equations by the Square Root Property • Square Root Property of Equations: If k is a positive number and if.
11.1 Solving Quadratic Equations
by the Square Root Property
• Square Root Property of Equations:
If k is a positive number and if a2 = k, then
a k
or
a- k
and the solution set is: {
k ,- k}
11.1 Solving Quadratic Equations
by the Square Root Property
• Example:
5 x 2
2
3
5 x 2 3 or 5 x 2 3
5 x 2 3 or 5 x 2 3
2 3
2 3
x
or x
5
5
11.2 Solving Quadratic Equations
by Completing the Square
• Example of completing the square:
x 6 x 7 0 cannotbe factored
2
x 6 x 9 2 0 (completethesquare)
2
( x 3) 2 0 ( x 3) 2
2
2
x 3 2 x 3 2
11.2 Solving Quadratic Equations
by Completing the Square
•
Completing the Square (ax2 + bx + c = 0):
1. Divide by a on both sides
(lead coefficient = 1)
2. Put variables on one side, constants on the
other.
3. Complete the square (take ½ of x coefficient
and square it – add this number to both sides)
4. Solve by applying the square root property
11.2 Solving Quadratic Equations
by Completing the Square
2
2
x
y
( x y )(x y )
• Review:
x2 y2
(prime)
x 3 y 3 ( x y )(x 2 xy y 2 )
x y ( x y )(x xy y )
3
3
2
2
x y ( x y )(x y ) ( x y)(x y)(x y )
4
4
2
2
2
2
2
• x4 + y4 – can be factored by completing the square
2
11.2 Solving Quadratic Equations
by Completing the Square
• Example:
Complete
the square:
y
x 2 x y y 2 x y
x y 2 xy
x y x
4
4
2 2
2 2
2
2 2
2
2
x y 2xy x y 2xy
2
2
2
2 2
2
Factor the difference
of two squares: 2
2 2
2
2
11.3 Solving Quadratic Equations
by the Quadratic Formula
• Solving ax2 + bx + c = 0:
Dividing by a: x2 b x c 0
a
a
Subtract c/a:
x ba x ac
2
Completing the
2
square by
x
adding b2/4a2:
b
a
x
b2
4a 2
c
a
b2
4a 2
11.3 Solving Quadratic Equations
by the Quadratic Formula
• Solving ax2 + bx + c = 0 (continued):
2
Write as a
2
b
4ac
b 2
4 ac
b
square: x 2 a 4 a 2 4 a 2 4a 2
Use square root
property:
b b2 4ac
x
2a
2a
b b2 4ac
Quadratic formula: x
2a
11.3 Solving Quadratic Equations
by the Quadratic Formula
• Quadratic Formula:
b b 4ac
x
2a
2
b 4ac
2
is called the discriminant.
If the discriminant is positive, the solutions are
real
If the discriminant is negative, the solutions are
imaginary
11.3 Solving Quadratic Equations
by the Quadratic Formula
• Example:
x2 5x 6 0
x
a 1, b -5, c 6
(5)
5
( 5) 4(1)(6)
2(1)
2
25 24 5 1
2
2 2
x 3, x 2
11.3 Solving Quadratic Equations
by the Quadratic Formula
•
Complex Numbers and the Quadratic Formula
Solve x2 – 2x + 2 = 0
(2) (2) 4(1)(2)
x
2(1)
2
2 4 2 4i 2 2i
2
2
2
1 i
11.4 Equations Quadratic in Form
Method
Advantages
Factoring
Fastest method
Disadvantages
Not always
factorable
2
Square root
Not always this
form : ( x a) b
property
form
Completing the Can always be Requires a lot
square
used
of steps
Quadratic
Can always be Slower than
Formula
used
factoring
11.4 Equations Quadratic in Form
•
Sometimes a radical equation leads to a quadratic
equation after squaring both sides
•
An equation is said to be in “quadratic form” if it
can be written as a[f(x)]2 + b[f(x)] + c = 0
Solve it by letting u = f(x); solve for u; then use
your answers for u to solve for x
11.4 Equations Quadratic in Form
•
Example:
4x
x 4x 3 x
4
2
2 2
2
3 0
Let u = x2
x
4 x 3 0 u 4u 3 0
(u 3)(u 1) 0 u 3, u 1
2 2
2
2
x 2 3, x 2 1 x 3 , x 1
11.5 Formulas and Applications
•
Example (solving for a variable involving a
square root)
4A
Solve : d
for A
d
2
4A
square bot h sides
d 2 4 A mult iplybot h sides by
d 2
4
A divide bot h sides by 4
11.5 Formulas and Applications
•
Example:
s 2t 2 kt
solvefor t
0 2t 2 kt s
get zero on right side
-k k 2 4( 2)( s )
t
2(2)
(quadrat ic formula)
-k k 2 8s
4
-k k 2 8s
-k k 2 8s
so t
and t
4
4
11.6 Graphs of Quadratic Functions
•
•
A quadratic function is a function that can be
written in the form:
f(x) = ax2 + bx + c
The graph of a quadratic function is a parabola.
The vertex is the lowest point (or highest point if
the parabola is inverted
11.6 Graphs of Quadratic Functions
•
Vertical Shifts:
f ( x) x k
2
The parabola is shifted upward by k units or
downward if k < 0. The vertex is (0, k)
•
Horizontal shifts:
f ( x) x h
2
The parabola is shifted h units to the right if h >
0 or to the left if h < 0. The vertex is at (h, 0)
11.6 Graphs of Quadratic Functions
•
Horizontal and Vertical shifts:
f ( x) x h k
2
The parabola is shifted upward by k units or
downward if k < 0. The parabola is shifted h
units to the right if h > 0 or to the left if h < 0
The vertex is (h, k)
11.6 Graphs of Quadratic Functions
•
Graphing:
f ( x) ax h k
2
1. The vertex is (h, k).
2. If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward (flipped).
3. The graph is wider (flattened) if 0 a 1
The graph is narrower (stretched) if a 1
11.6 Graphs of Quadratic Functions
Inverted Parabola with Vertex (h, k)
f ( x) x h k
2
Vertex = (h, k)
11.7 More About Parabolas;
Applications
•
Vertex Formula:
The graph of f(x) = ax2 + bx + c has vertex
b
,
2a
b
f
2a
11.7 More About Parabolas;
Applications
•
Graphing a Quadratic Function:
1. Find the y-intercept (evaluate f(0))
2. Find the x-intercepts (by solving f(x) = 0)
3. Find the vertex (by using the formula or by
completing the square)
4. Complete the graph (plot additional points as
needed)
11.7 More About Parabolas;
Applications
•
Graph of a horizontal (sideways) parabola:
The graph of x = ay2 + by + c or x = a(y - k)2 + h
is a parabola with vertex at (h, k) and the
horizontal line y = k as axis. The graph opens to
the right if a > 0 or to the left if a < 0.
11.7 More About Parabolas
Horizontal Parabola with Vertex (h, k)
x y k h
2
Vertex = (h, k)