Lesson 1.4 Solving Quadratic Equations
Objective: Day 1: To solve quadratic equations by factoring and completing the square.
Quadratic Equation: ax2 + bx + c =0
A root, solution, or zeroes of the equation are values that satisfy the equation.
· FACTORING: Find the roots of the following
a. 9x2 – 4 b. x2 + 6x + 9
c. x2 – 7x + 12 d. 2x2 +x-15
e. 3x2 + 10x – 8 = -11
Solve: (2x-1)2 = 4
· Completing the Square: Transforming the quadratic so that one side is a perfect square trinomial
a. y2 – 8y = 2 b. 2x2 – 12x – 7 = 0
c. x2 + 14x = 374 d. 4x2 – 4x – 17 = 0
p. 118 #8-16 even #26-44 even
Lesson 1.4 Solving Quadratic Equations Continued….
Objective: Day 2: To solve quadratic equations by using the quadratic formula, and identify the best method of solving an equation.
Do now: Solve the following using the quadratic formula.
a. 2x2 + 7 = 4x b. 2x2 – 12x – 7 = 0
Discriminant: b2-4ac
If….. b2-4ac > 0
b2-4ac < 0
b2-4ac = 0
Example: Use the discriminant to determine the number and type of solutions of the following
a. 2x2 – 5x + 5 = 0 b. 2x2 – x – 1 = 0
Identify the easiest method for solving the following equations.
1. x2 + 14x = 374 2. 4x2 – 5x = 0 3. 3x3 + 11x -4=0
4. 4x2 – x – 7 = 0 5. px2 + qx + r = 0
Solve by whichever method seems easiest.
a. 8x2 = 7 – 10x b. (3x-2)2 = 121
c. (4x+7)(x-1) = 2(x-1) d. x+3x-3+x-3x+3=18-6xx2-9
e. t2+1t+2=t3+5t+2 f. 2x=x-8
p. 118 #60-74 even #100-106 even
Lesson 1.5 The Complex numbers
Objective: Students will add, subtract, multiply, and divide complex numbers.
DO NOW: Simplify the following radicals:
a. 20 b. -75x2 c. 125y3
d. 5+23 e. 32-5 f. -16
Imaginary unit i: i =
Ex. Simplify a. -25 b. -7
Complex Numbers: any number of the form a +bi, where a and b are real numbers and I is the imaginary unit.
Ex.
a is the real part and b is called the imaginary part.
Pure imaginary numbers:
Two complex numbers are equal only if their real parts are equal and their imaginary parts are equal.
Powers of i
i = i5 = i9=
i2 = i6 = i10=
i3 = i7= i11=
i4 = i5 = i12=
Ex. Simplify the following
a. i100 b. i81 c. i23 + i54
Operations on complex numbers are done by treating i as a variable.
Ex. (2+3i) + (4+5i) Ex. (5-6i) – (3 + 2i)
Ex. (2+3i)(4+5i) Ex. (5-2i)(5+2i)
Complex Conjugates:
Ex. Simplify the following
a. 2i-34i b. 5-2i4+3i
HW: Worksheet
Lesson 1.6 Special cases of Solving Quadratics and Applications
Objective: Students will be able to solve special cases of quadratic equations and will also be able to apply solving techniques to real world applications.
DO NOW: Simplify the following
a. 3+2i2i b. 2-3i4+i
Try the following….. Solve for x:
a. b.
c.
Quadratic Equations Word Problem Applications
EXAMPLES:
1. When the square of a certain number is diminished by 9 times the number the result is 36. Find the number.
2. Find two consecutive positive integers such that the square of the first is decreased by 17 equals 4 times the second.
3. The ages of three family children can be expressed as consecutive integers. The square of the age of the youngest child is 4 more than eight times the age of the oldest child. Find the ages of the three children.
4. The altitude of a triangle is 5 less than its base. The area of the triangle is 42 square inches. Find its base and altitude.
5. If the measure of one side of a square is increased by 2 centimeters and the measure of the adjacent side is decreased by 2 centimeters, the area of the resulting rectangle is 32 square centimeters. Find the measure of one side of the square.
6. Joe’s rectangular garden is 6 meters long and 4 meters wide. He wishes to double the area of his garden by increasing its length and width by the same amount. Find the number of meters by which each dimension must be increased.
7. If the length of one side of a square is tripled and the length of an adjacent side is increased by 10, the resulting rectangle has an area that is 6 times the area of the original square. Find the length of a side of the original square.
Lesson 1.7 Graphing Quadratic Functions
Objective: Students will be able to graph quadratic functions in standard form. They will also be able to describe important characteristics of the graph such as vertex, max/min, x intercepts and y intercept.
DO NOW: Find the x-intercepts of the following quadratic functions.
a. f(x) = x2 – 5x + 6 b. f(x) = 3x2 +2x + 4
· The Quadratic Equation is written as: ______, this equation has a degree of ______.
§ Where a, b and c are integer coefficients (where a 0)
· The graph of this equation is called a ______, it is ______
Draw in the line of symmetry of the parabola on the grid.
This line is called the ______
· It is always a vertical line that goes through the turning point of the curve.
Formula: Axis of Symmetry:
Vertex: The turning point on the graph. The “vertex” has the coordinates of .
To Find The Vertex
Roots of the equation are the points where the parabola
______the x – axis, so y = ______,
What are the roots of the parabola on the grid to the left? ______
GRAPHING QUADRATIC FUNCTIONS
How to Graph Parabolas:
1. Find the axis of symmetry by using the formula.
2. Substitute the x value back into the equation to find the vertex.
3. Determine if the parabola has a maximum or minimum value.
4. Find the y-intercept.
3. Make a table of values. Your graph should clearly show 5 points. Remember, the graph the symmetric so you can reflect some of the points.
4. Graph the points.
EX1: GRAPH: + 2
EX2: GRAPH: