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Solving Quadratic Equations
What is a quadratic equation ?
The standard form of a quadratic equation is ax ^{2} + bx + c = 0 , where
a ≠ 0, b and
c are real numbers
Examples:
An equation ab=0 is true if and only if a=0 or b=0, or both , (A
product is o if and only if at least one factor is o.)
Ex. 12x^{2} − 5x − 2 = 0
Ex. Solve
x^{2} = 9
x^{2} = 8
x^{2} = −16
Ex. Solve
x^{2} − 3 = 2
2x^{2} = 36
2x^{2} − 5 = −4
4x^{2} +15 = 3
For any real number k and any algebraic espression x,
If x^{2}=k, then
Ex. Solve…
(x − 3)^{2} = 25
(x + 2)^{2} = 4
(x +1)^{2} = 12
(x − 5)^{2} = −9
(5x − 3)^{2} = 8
Ex. Given f (x) = x^{2} − 6x + 2
a. Find f (−3)
b. Where is f (x) = −3
Ex. Given the function f (x) = x^{2} − 6x + 2
a. Find the yintercept
b. Find the xintercepts
Completing the Square
Ex. Solve x ^{2} −10x + 25 = 100
How to complete the square:
1. Start with x^{2} + bx
2. Evaluate
3. Evaluate
4. Combine
5. Factor, end up with
Ex. Complete the square
x^{2} − 8x
x^{2} +10x
x^{2} −5x
x^{2} + 7x
1. If necessary, move c to the righthand side of the equation
2. If necessary, divide both sides of the equation by a
3. Complete the square on the lefthand side
4. Balance the equation by adding to righthand side.
5. Factor and solve using Principal of Square Roots
Ex. Solve by completing the square
x^{2} − 6x +1 = 15
x^{2} +10x − 7 = 9
x^{2} − 7x −3 =10
3x^{2} −12x −8 = 25
Section 11.2 Quadratic Formula
Developing the formula…
Ex. Solve…
2x^{2} −9x + 5 = 0
Ex. Solve…
4x^{2} = 3+ 7x
3x^{2} + 2x = 7
12x^{2} − 5x − 2 = 0
Ex. Given where is f (x) = 1?
Ex. Given f(x) = 2x^{2}5x+1
a. Find the yintercept
b. Find the xintercept(s), if they exist.
Which method to use?
Factoring
Principal of Square Roots
Completing the Square
Quadratic Formula
Section 11.3 Formulas and Applications
Ex. Solve for d:
Ex. Solve for t: Y = rt ^{2} − st
Ex. Sandi’s Subaru travels 280 mi averaging a certain speed. If the car had g one
5
mph faster, the trip would have taken 1 hr less. Find Sandi’s average speed.
Distance  Rate  Time  
Ex. A lot is in the shape of a right triangle. The shorter leg measures 120 m.
The
hypotenuse is 40 m longer than the length of the longer leg. How long is the
longer leg?
Ex. The position of an object moving in a straight line is given by s(t) = −t ^{
2}
+ 8t ,
where s is in feet and t is the time in seconds the object has been in motion.
How long will it take the object to move 13 ft?
Section 11.6 Graphing Quadratic Functions
What is a quadratic function? What does its graph look like ?
• General Form
• Standard Form
The graph of a quadratic function/parabola
o Vertex  
o Axis (or Line) of Symmetry  
o Curvature  
o Intercepts  
o Min/Max Value 
Ex. Graph f (x) = 2x^{2} − 3
Vertex:  
Axis of Symmetry Curvature 

Min/Max Value 
Ex. Graph f (x) = −2(x + 3)^{2}
Ex. Graph f (x) = 3(x − 2)^{2} +1
Ex. Graph f (x) = −4(x + 3)^{2} − 2
Ex. Graph
Ex. Graph
Section 11.7 More Quadratic Functions
General Form of Quadratic Function
Standard/Vertex Form of Quadratic Function
How do you transform a quadratic function to standard form?
Ex. Write in standard form. f (x) = x^{2} + 6x
Ex. Write in standard form. f (x) = x^{2} + 4x + 3
Ex. Write in standard form. f (x) = x^{2} − 8x + 23
Ex. Write in standard form. f (x) = 2x^{2} +12x + 3
Ex. Graph the function by first finding the vertex, yintercepts, xintercepts
(if
they exist) and any information about the shape
f (x) = 3x^{2} − 24x + 43
Ex. Graph the function by first finding the vertex, yintercepts, xintercepts
(if
they exist) and any information about the shape
f (x) = −4x^{2} + 8x −1
The vertex of the parabola given by f(x)=ax^{2}+bx+c is
The x coordinate of the vertex is b/(2a). The axis of symmetry is
x= b/(2a). The second coordinate of the vertex is most
commonly found by computing
Section 11.8 Maximum and Minimum Problems
We have seen that for any quadratic function f, the value of f (x) at the vertex
is either a
maximum or a minimum. Thus problems in which a quantity must be maximized or
minimized can be solved by finding the coordinates of the vertex, assuming the
problem
can be modeled with a quadratic function.
f(x) at the vertex a minimum
f(x) at the vertex a maximum
Ex. Find the maximum or minimum value of the function f(x)=x^{2}10x+21
Ex. Find the maximum or minimum value of the function f(x)= 3x^{2}+6x+2
Ex. The value of a share of I. J. Solar can be represented by V(x)=x^{2}6x+13,
where x is the number of months after January 2004. What is the lowest value
V(x)
will reach, and when did that occur?
Ex. Recall that total profit P is the difference between total revenue R and
total cost C.
Given R(x) = 1000x x^{2} and C(x) = 3000 + 2x, find the total profit,
the
maximum value of the total profit, and the value of x at which it occurs.
Ex. The perimeter of a rectangle is 40 inches. Determine the dimensions that
maximize the area of the rectangle.
Ex. A farmer has 200 ft of fence with which to form a rectangular pen on his
farm. If
an existing fence forms one side of the rectangle, what dimensions will maximize
the size of the area?
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