9.1 Square Roots and the Pythagorean Theorem
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Transcript of 9.1 Square Roots and the Pythagorean Theorem
By definition 25 is the number you would multiply times itself to get 25 for an answer.
Because we are familiar with multiplication, we know that 25 = 5
Numbers like 25, which have whole numbers for their square roots, are called perfect squares
You need to memorize at least the first 15 perfect squares
9.1 Square Roots and the Pythagorean Theorem
Perfect square
Square root
1 1 = 1
4 4 = 2
9 9 = 3
16 16 = 4
25 25 = 5
36 36 = 6
49 49 = 7
64 64 = 8
81 81 = 9
100 100 = 10
121 121 = 11
144 144 = 12
169 169 = 13
196 196 = 14
225 225 = 15
Perfect square
Square root
Every whole number has a square root
Most numbers are not perfect squares, and so their square roots are not whole numbers.Most numbers that are not perfect squares have square roots that are irrational numbers
Irrational numbers can be represented by decimals that do not terminate and do not repeat
The decimal approximations of whole numbers can be determined using a calculator
9.1 Square Roots and the Pythagorean Theorem
Find the square roots of the given numbersIf the number is not a perfect square, use a calculator to find the answer correct to the nearest hundredth.
81
37
158
9.1 Square Roots and the Pythagorean Theorem
981
08.637
57.12158
Find the square roots of the given numbersIf the number is not a perfect square, use a calculator to find the answer correct to the nearest thousandth.
9.1 Square Roots and the Pythagorean Theorem
12144
3625
65
04. 2.
49 undefined
49 7
9.1 Square Roots and the Pythagorean Theorem
The Pythagorean TheoremFor any right triangle, the sum of the squares of the lengths of the legs a and b, equals the square of the length of the hypotenuse.
a2 + b2 = c2
a
b
c
9.1 Square Roots and the Pythagorean Theorem
Find c. a2 + b2 = c2 6
8
c
222 86 c2100 c
100c10c
9.1 Square Roots and the Pythagorean Theorem
Find c. a2 + b2 = c2 4
6
c
222 64 c
522 c
52c21.7c
9.1 Square Roots and the Pythagorean Theorem
Find a. a2 + b2 = c2 a
8
17
222 178 a
2252 a225a
15a
289642 a
9.1 Square Roots and the Pythagorean Theorem
The length of each side of a softball field is 60 feet. How far is it from home to second?
602 + 602 = c2
3600 + 3600 = 7200c2 = 7200
60 ft
60 ft
ftc 85.847200
9.2 Solving Quadratic Equations
Solving x2 = d by Finding Square Roots1. If d is positive, then x2 = d has two solutions2. The equation x2 = 0 has one solutions:3. If d is negative, then x2 = d has no solution.
9.2 Solving Quadratic Equations
Solve 1. x2 = 492. x2 = 123. x2 = 04. x2 = -9
7x46.3x
0xPossibleNot
9.2 Solving Quadratic Equations
Solve 1. 3x2 +1 = 762. 4x2 + 6 = 703. 5x2 – 7 = 184. 3x2 – 10 = 38
5x4x
23.2x4x
9.3 Graphing Quadratic Equations
y = ax2 + bx + c
The graph of a quadratic function is a parabola.
A parabola can open up or down.
If the parabola opens up, the lowest point is called the vertex.
If the parabola opens down, the vertex is the highest point.
NOTE: if the parabola opened left or right it would not be a function!
y
x
Vertex
Vertex
9.3 Graphing Quadratic Equations
y = ax2 + bx + c
The parabola will open down when the a value is negative.
The parabola will open up when the a value is positive.
y
x
The standard form of a quadratic function is
a > 0
a < 0
9.3 Graphing Quadratic Equations
y
x
Line of SymmetryParabolas have a symmetric
property to them.
If we drew a line down the middle of the parabola, we could fold the parabola in half.
We call this line the line of symmetry.
The line of symmetry ALWAYS passes through the vertex.
Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side.
9.3 Graphing Quadratic Equations
Find the line of symmetry of y = 3x2 – 18x + 7
When a quadratic function is in standard form
The equation of the line of symmetry is
y = ax2 + bx + c,
2ba
x
For example…
Using the formula…
This is best read as …
the opposite of b divided by the quantity of 2 times a.
18
2 3x 18
6 3
Thus, the line of symmetry is x = 3.
9.3 Graphing Quadratic Equations
We know the line of symmetry always goes through the vertex.Thus, the line of symmetry gives us the x – coordinate of the vertex.
To find the y – coordinate of the vertex, we need to plug the x – value into the original equation.
STEP 1: Find the line of symmetry
STEP 2: Plug the x – value into the original equation to find the y value.
y = –2x2 + 8x –3
8 8 22 2( 2) 4
ba
x
y = –2(2)2 + 8(2) –3
y = –2(4)+ 8(2) –3
y = –8+ 16 –3 y = 5
Therefore, the vertex is (2 , 5)
9.3 Graphing Quadratic Equations
The standard form of a quadratic function is given by y = ax2 + bx + c
STEP 1: Find the line of symmetry
STEP 2: Find the vertex
STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.
9.3 Graphing Quadratic Equations
STEP 1: Find the line of symmetry
Let's Graph ONE! Try …
y = 2x2 – 4x – 1
( )4 1
2 2 2bxa
-= = =
y
x
Thus the line of symmetry is x = 1
9.3 Graphing Quadratic Equations
Let's Graph ONE! Try …
y = 2x2 – 4x – 1
STEP 2: Find the vertex
y
x
( ) ( )22 1 4 1 1 3y = - - =- Thus the vertex is (1 ,–3).
Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex.
9.3 Graphing Quadratic Equations
Let's Graph ONE! Try …
y = 2x2 – 4x – 1
( ) ( )22 3 4 3 1 5y = - - =
STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.
y
x( ) ( )22 2 4 2 1 1y = - - =-
9.3 Graphing Quadratic Equations
x y2 -13 5
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
Root (-4., 0.) Root (2., 0.)
Vertex (-1., 9.)
9.3 Graphs of Quadratic Equations
-Vertex:x =(-b/2a)x= -(-2/2(-1))x= 2/(-2)x= -1
Solve for y:
y = -x2 -2x + 8
y = -(-1)2 -(2)(-1) + 8
y = -(1) + 2 + 8
y = 9Vertex is (-1, 9)
For y = -x2 - 2x + 8 identify each term, graph the equation, find the vertex, and find the solutions of the equation.
The quadratic formula allows you to find the roots of a quadratic equation (if they exist) even if the quadratic equation does not factorise.
The formula states that for a quadratic equation of the form :
ax2 + bx + c = 0
The roots of the quadratic equation are given by :
aacbbx
242
9.4 The Quadratic Formula
The Quadratic Formula.
aacbbx
242
Example 1
Use the quadratic formula to solve the equation :
x 2 + 5x + 6 = 0
Solution:
x2 + 5x + 6= 0
a = 1 b = 5 c = 6
aacbbx
242
12)614(55 2
x
2)24(255
x
215
x
215
215
xorx
x = - 2 or x = - 3
These are the roots of the equation.
Example 3
Use the quadratic formula to solve the equation :
8x2 - 22x + 15= 0
Solution:
8x2 - 22x + 15= 0
a = 8 b = -22 c = 15
aacbbx
242
82)1584()22()22( 2
x
16)480(484(22
x
16422
x
16222
16222
xorx
x = 3/2 or x = 5/4
These are the roots of the equation.
Example 4
Use the quadratic formula to solve for x to 2 decimal places.
2x2 + 3x - 7= 0
Solution:
2x 2 + 3x – 7 = 0
a = 2 b = 3 c = - 7
aacbbx
242
22)724(33 2
x
4)56(93
x
4653
x
40622.83
40622.83
xorx
x = 1.27 or x = - 2.77
These are the roots of the equation.
9.5 Problem Solving Using the Discriminant
aacbbx
242
Discriminant
9.5 Problem Solving Using the Discriminant
The number of solutions in a quadratic equationConsider the equation ax2 + bx + c = 0
1. If b2 – 4ac > 0, then the equation has 2 solutions.2. If b2 – 4ac = 0, then the equation has 1 solution.3. If b2 – 4ac < 0, then the equation has no solution.
Find the discriminant of 3x2 + x – 2 = 0 and tell the nature of its roots.
Discriminant = b2 – 4ac = 12 – 4(3)(-2) = 1 – (-24) = 1 + 24
= 25
So, there are two solutions
9.5 Problem Solving Using the Discriminant
9.5 Problem Solving Using the Discriminant
Determine the number of solutions
a. 2x2 – x + 3 = 0b. x2 + x + 4 = 0c. 3x2 –5x - 3 = 0d. 2x2 – x - 9 = 0
9.5 Problem Solving Using the Discriminant
Match the discriminant with the graph
a. b2 – 4ac = 7 b. b2 – 4ac = -2 c. b2 – 4ac = 0
9.6 Graphing Quadratic Inequalities
1. Draw the graph of the equation obtained by replacing the inequality sign by an equal sign. Use a dashed line if the inequality is < or >. Use a solid line if the inequality is ≤ or ≥.
2. Check a point in each of the two regions of the plane determined by the graph of the equation. If the point satisfies the inequality, then shade the region containing the point.
Steps for Drawing the Graph of an Inequality in Two Variables
9.6 Graphing Quadratic Inequalities
Solve y > x2