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Martin-Gay, Developmental Mathematics 1
Warm Up
2(3 4)(3 4)x x
218 32x Factor the following
Solving Quadratic Equations by the Square
Root Property
Martin-Gay, Developmental Mathematics 3
Square Root Property
We previously have used factoring to solve quadratic equations.
This chapter will introduce additional methods for solving quadratic equations.
Square Root PropertyIf b is a real number and a2 = b, then
ba
Martin-Gay, Developmental Mathematics 4
Solve x2 = 49
2x
Solve (y – 3)2 = 4
Solve 2x2 = 4
x2 = 2
749 x
y = 3 2
y = 1 or 5
243 y
Square Root Property
Example
Martin-Gay, Developmental Mathematics 5
Solve x2 + 4 = 0 x2 = 4
There is no real solution because the square root of 4 is not a real number.
Square Root Property
Example
Martin-Gay, Developmental Mathematics 6
Solve (x + 2)2 = 25
x = 2 ± 5
x = 2 + 5 or x = 2 – 5
x = 3 or x = 7
5252 x
Square Root Property
Example
Martin-Gay, Developmental Mathematics 7
Solve (3x – 17)2 = 28
72173 x
3
7217 x
7228 3x – 17 =
Square Root Property
Example
Solving Quadratic Equations by the
Quadratic Formula
https://www.youtube.com/watch?v=YCuXiujC3KE
Martin-Gay, Developmental Mathematics 9
The Quadratic Formula
Another technique for solving quadratic equations is to use the quadratic formula.
The formula is derived from completing the square of a general quadratic equation.
Martin-Gay, Developmental Mathematics 10
Quadratic Formula
• If we are unable to factor a quadratic function to find the roots we can utilize the Quadratic Formula. The entire equation can tell us the number of roots & the radicand tells us the number of real solutions. A quadratic equation written in standard form, ax2 + bx + c = 0, has the solutions
Martin-Gay, Developmental Mathematics 11
Discriminant
Positive, perfect squares 2 real, rational roots
Positive, not perfect squares 2 real, irrational roots
Zero 1 real rational root
Negative 2 complex roots
Martin-Gay, Developmental Mathematics 12
Martin-Gay, Developmental Mathematics 13
Martin-Gay, Developmental Mathematics 14
Martin-Gay, Developmental Mathematics 15
Martin-Gay, Developmental Mathematics 16
Solution
Martin-Gay, Developmental Mathematics 17
Martin-Gay, Developmental Mathematics 18
Martin-Gay, Developmental Mathematics 19
Solve 11n2 – 9n = 1 by the quadratic formula.
11n2 – 9n – 1 = 0, so
a = 11, b = -9, c = -1
)11(2
)1)(11(4)9(9 2
n
22
44819
22
1259
22
559
The Quadratic Formula
Example
Martin-Gay, Developmental Mathematics 20
)1(2
)20)(1(4)8(8 2
x
2
80648
2
1448
2
128 20 4 or , 10 or 22 2
x2 + 8x – 20 = 0 (multiply both sides by 8)
a = 1, b = 8, c = 20
8
1
2
5Solve x2 + x – = 0 by the quadratic formula.
The Quadratic Formula
Example
Martin-Gay, Developmental Mathematics 21
Solve x(x + 6) = 30 by the quadratic formula.
x2 + 6x + 30 = 0
a = 1, b = 6, c = 30
)1(2
)30)(1(4)6(6 2
x
2
120366
2
846
So there is no real solution.
The Quadratic Formula
Example
Martin-Gay, Developmental Mathematics 22
The expression under the radical sign in the formula (b2 – 4ac) is called the discriminant.
The discriminant will take on a value that is positive, 0, or negative.
The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively.
The Discriminant
Martin-Gay, Developmental Mathematics 23
Use the discriminant to determine the number and type of solutions for the following equation.
5 – 4x + 12x2 = 0
a = 12, b = –4, and c = 5
b2 – 4ac = (–4)2 – 4(12)(5)
= 16 – 240
= –224
There are no real solutions.
The Discriminant
Example
Martin-Gay, Developmental Mathematics 24
Solving Quadratic Equations
Steps in Solving Quadratic Equations1) If the equation is in the form (ax+b)2 = c, use
the square root property to solve.
2) If not solved in step 1, write the equation in standard form.
3) Try to solve by factoring.
4) If you haven’t solved it yet, use the quadratic formula.
Martin-Gay, Developmental Mathematics 25
Solve 12x = 4x2 + 4.
0 = 4x2 – 12x + 4
0 = 4(x2 – 3x + 1)
Let a = 1, b = -3, c = 1
)1(2
)1)(1(4)3(3 2
x
2
493
2
53
Solving Equations
Example
Martin-Gay, Developmental Mathematics 26
Solve the following quadratic equation.
02
1
8
5 2 mm
0485 2 mm
0)2)(25( mm
02025 mm or
25
2 mm or
Solving Equations
Example
§ 16.4
Graphing Quadratic Equations in Two
Variables
Martin-Gay, Developmental Mathematics 28
We spent a lot of time graphing linear equations in chapter 3.
The graph of a quadratic equation is a parabola.
The highest point or lowest point on the parabola is the vertex.
Axis of symmetry is the line that runs through the vertex and through the middle of the parabola.
Graphs of Quadratic Equations
Martin-Gay, Developmental Mathematics 29
x
y
Graph y = 2x2 – 4.
x y
0 –4
1 –2
–1 –2
2 4
–2 4
(2, 4)(–2, 4)
(1, –2)(–1, – 2)
(0, –4)
Graphs of Quadratic Equations
Example
Martin-Gay, Developmental Mathematics 30
Although we can simply plot points, it is helpful to know some information about the parabola we will be graphing prior to finding individual points.
To find x-intercepts of the parabola, let y = 0 and solve for x.
To find y-intercepts of the parabola, let x = 0 and solve for y.
Intercepts of the Parabola
Martin-Gay, Developmental Mathematics 31
If the quadratic equation is written in standard form, y = ax2 + bx + c,
1) the parabola opens up when a > 0 and opens down when a < 0.
2) the x-coordinate of the vertex is . a
b
2
To find the corresponding y-coordinate, you substitute the x-coordinate into the equation and evaluate for y.
Characteristics of the Parabola
Martin-Gay, Developmental Mathematics 32
x
yGraph y = –2x2 + 4x + 5.
x y
1 7
2 5
0 5
3 –1
–1 –1
(3, –1)(–1, –1)
(2, 5)(0, 5)
(1, 7)Since a = –2 and b = 4, the graph opens down and the x-coordinate of the vertex is 1
)2(2
4
Graphs of Quadratic Equations
Example
§ 16.5
Interval Notation, Finding Domain and Ranges from
Graphs, and Graphing Piecewise-Defined Functions
Martin-Gay, Developmental Mathematics 34
Recall that a set of ordered pairs is also called a relation.
The domain is the set of x-coordinates of the ordered pairs.
The range is the set of y-coordinates of the ordered pairs.
Domain and Range
Martin-Gay, Developmental Mathematics 35
Find the domain and range of the relation {(4,9), (–4,9), (2,3), (10, –5)}
• Domain is the set of all x-values, {4, –4, 2, 10}
• Range is the set of all y-values, {9, 3, –5}
Example
Domain and Range
Martin-Gay, Developmental Mathematics 36
Find the domain and range of the function graphed to the right. Use interval notation. x
y
Domain is [–3, 4]
Domain
Range is [–4, 2]
Range
Example
Domain and Range
Martin-Gay, Developmental Mathematics 37
Find the domain and range of the function graphed to the right. Use interval notation. x
y
Domain is (– , )
DomainRange is [– 2, )
Range
Example
Domain and Range
Martin-Gay, Developmental Mathematics 38
Input (Animal)• Polar Bear• Cow• Chimpanzee• Giraffe• Gorilla• Kangaroo• Red Fox
Output (Life Span)
20
15
10
7
Find the domain and range of the following relation.
Example
Domain and Range
Martin-Gay, Developmental Mathematics 39
Domain is {Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fox}
Range is {20, 15, 10, 7}
Domain and Range
Example continued
Martin-Gay, Developmental Mathematics 40
Graph each “piece” separately.
Graph3 2 if 0
( ) . 3 if 0
x xf x
x x
Graphing Piecewise-Defined Functions
Example
Continued.
x f (x) = 3x – 1
0 – 1(closed circle)
–1 – 4
–2 – 7
x f (x) = x + 3
1 4
2 5
3 6
Values 0. Values > 0.
Martin-Gay, Developmental Mathematics 41
Example continued
Graphing Piecewise-Defined Functions
x
y
x f (x) = x + 3
1 4
2 5
3 6
x f (x) = 3x – 1
0 – 1(closed circle)
–1 – 4
–2 – 7
(0, –1)
(–1, 4)
(–2, 7)
Open circle (0, 3)
(3, 6)