4-6 Perform Operations with Complex NumbersName:______
Objective: To use complex numbers to perform operations.
Algebra 2 Standards 5.0, 6.0, and 8.0
* Sometimes a quadratic equation does not have a real-number solution(Ex: x2 = -1).To overcome this, there is an expanded system of numbers using the imaginary unit (i),
defined as ______.
The imaginary unit i can be used to write the square root of any negative number.
*The Square Root of a Negative Number
Property / Example- If r is a positive real number, then
- By Property (1), it follows that
Ex. 1: Solve You Try: Solve
a. x2 = -13 a. x2 = -38
b. 2x2 + 18 = -72 b. 3x2 – 7 = -31
*A Complex Number written in standard form is a number ______where a and b are real numbers. The number a is the real part of the complex number and the number bi is the imaginary part.Complex Numbers
Real Numbers
(a + 0i)
-1
/ Imaginary Numbers
(a + bi, )
2 + 3i 5 – 5i
Pure Imaginary Numbers
(0 + bi, )
-4i 6i
If then a + bi is an imaginary number. If a = 0 and then a + bi is a pure imaginary number.
Two complex numbers a + bi and c + di are equal if and only if a = c and b = d.
*Sums and Differences of Complex Numbers: To add or subtract two complex numbers, add or subtract their real parts and their imaginary parts separately.
Sum of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i
Difference of complex numbers:(a + bi) – (c + di) = (a – c) + (b – d)i
Ex. 2: Write the expression as a complex number in standard form.
a. (12 – 11i) + (-8 + 3i)b. (15 – 9i) – (24 – 9i)c. 35 – (13 + 4i) + i
You Try: Write the expression as a complex number in standard form.
a. (9 – i) + (-6 + 7i) b. (3 + 7i) – (8 – 2i)c. -4 – (1 + i) – (5 + 9i)
*Multiplying Complex Numbers:To multiply two complex numbers, use the Distributive Property or the FOIL method.
Ex. 3: Write the expression as a complex number in standard form.
a.-5i(8 – 9i)b.(-8 + 2i)(4 – 7i) c.
You Try: Write the expression as a complex number in standard form.
a.i(7-3i)b.(6 + i)(5- 3i) c.
*Complex Conjugates:Two complex numbers of the form a + bi and a – bi are called complex conjugates.
The product of complex conjugates is always a ______number.
We can use conjugates to write the quotient of two complex numbers in standard form.
Ex. 4. Write the expression in standard form.
a. b. c.
You Try: Write the expression in standard form.
a. b. c.
.
*Complex Plane: Every complex number corresponds to a point in the complex plane.The complex plane has a horizontal axis called the ______axis and a vertical axis called the ______axis
Ex. 5: Plot the complex numbers in the same complex plane.
a. 4 + 2i (start at the origin, move right 4 and up 2)
b. -1 + 3i
c. -4i
d. 2 – 2i
You Try: Plot the complex numbers in the same complex plane.
a. 4 – i
b. -3 – 4i
c. 2 – 5i
d. 3i
The absolute value of a complex number
z = a + bi, denoted ,
is a nonnegative real number defined as .
This is the ______
between z and the origin in the complex plane.
Ex. 6: Find the absolute value of each.
a.5 – 12i b.17i
You Try: Find the absolute value of each.
a.4 – ib. – 4i
c. 2 + 5id. –2 + 3i
4-7 Complete the SquareName:______
Objective: To solve quadratic equations by completing the square.
Algebra 2 Standards 8.0 and 10.0
*Recall: We can solve by taking the square root.If one side of an equation is a perfect square trinomial, we can also solve it by taking the square root.
Ex. 1: Solvex2 + 20x + 100 = -81You Try: Solve x2 – 10x + 25 = -1
*Completing the Square:In example 1, x2 + 20x + 100 is a perfect square trinomial because it equals (x + 10)2.
Sometimes you need to add a term to an expression ______to make it a square.
This process is called Completing the Square.
Words: To complete the square for the expression x2 + bx, add ______.
Algebra:
Steps: 1)Find half of x-term coefficient and square it :
2)Add and write as a binomial squared.
Ex. 2: Find the value of c that makes x2 – 28x + c a perfect square trinomial. Then write the expression as the square of a trinomial.
You Try: Find the value of c that makes x2 + 16x + c a perfect square trinomial. Then write the expression as the square of a trinomial.
*Solving equations: You can use completing the square to solve any quadratic equation. You must be sure to add the same number to both sides of the equation.*Solve ax2 + bx+ c = 0 when a = 1 using completing the square
Ex.) x2 – 10x + 1 = 0
1) Move constant to the right and leave space
2) Take , square it, and add that value to both sides
3) Factor left side as and simplify the right side.
4) Take square root both sides (!)
5) Solve for x
Example 3: Solve the equation by completing the square.
- b.
You try: Solve x2 – 12x + 8 = 0You Try:Solve
by completing the square.
*Solve ax2 + bx+ c = 0 when a ≠ 1To complete the square, you must divide everything by a so that
Ex. 4: Solve 3x2 – 36x + 150 = 0 by completing the square. You Try: Solve 2x2 – 4x – 14 = 0
Ex. 5: Solve 6x(x + 8) = 12 by completing the square.
*Vertex Form:Recall that the vertex form of a quadratic function is ______where (h, k) is the ______of the function’s graph. We can use completing the square to write a quadratic function in vertex form.
Ex. 6: Write y = x2 + 18x + 95 in vertex form. Then identify the vertex.
(steps)
- Write the original function
- Move constant to y side
- Add to each side
- Write trinomial as a binomial squared
- Solve for y
You Try: Write y = x2 – 8x + 17 in vertex form. Then identify the vertex.
Ex. 7: The height y (in feet) of a ball that was thrown up in the air from the roof of a building after t seconds is given by the equation y = -16t2 + 64t + 50. Find the maximum height of the ball.
You Try: Write y = 2x2 – 4x – 4 in vertex form. Then identify the vertex.
**Deriving the Quadratic Formula by Completing the Square
We will develop the Quadratic Formula by starting with the standard form of a quadratic equation and then solve for x by completing the square.
, where . Ex:
Step 1 / ,Step 2 /
Step 3 /
Step 4 /
Step 5 /
Step 6 /
Step 7 /
Step 8 /
Step 9 /
Step 10 /
Step 11 /
4-8 Use the Quadratic Formula and the Discriminant Name:______
Objective: To solve quadratic equations using the quadratic formula.
Algebra 2 Standard 8.0
*The Quadratic Formula: Let a, b, and c be real numbers such that a ≠ 0.The solutions of the quadratic equation ax2 + bx + c = 0 are
______.
---Be sure that the original equation is in standard form.
Ex. 1: Solve x2– 5x = 7You Try: Solve x2 = 6x – 4
Ex. 2: Solve 16x2 – 23x = 17x – 25 You Try: Solve 4x2 – 10x = 2x – 9
*In Algebra 1, if we got a negative number inside the radical, we said the equation had no real solutions. Now we will take the equation further using imaginary solutions.Ex. 3: Solve x2 – 6x + 10 = 0You Try: 7x – 5x2 – 4 = 2x +3
*Using the Discriminant of ax2 + bx + c = 0
The Quadratic Formula: , b2 – 4ac is called the Discriminant.Value of the discriminant / b2 – 4ac > 0 / b2 – 4ac = 0 / b2 – 4ac < 0
Number and type of solutions / ______real solutions / _____ real solution / Two ______solutions
Graph of
ax2 + bx + c = 0 /
______x-intercepts /
______x-intercept /
______x-intercepts
Ex. 4: Find the discriminant of the quadratic equation and give the number and type of solutions of the equation.
a. x2 + 10x + 23 = 0b. x2 + 10x + 25 = 0c. x2 + 10x + 27 = 0
You Try: Find the discriminant of the quadratic equation and give the number and type of solutions of the equation.
a. 3x2 + 12x + 12 = 0b. 2x2 + 4x – 4 = 0c. 7x2– 2x = 5
*Modeling Launched Objects: Recall that the formula for a dropped object is h = -16t2 + h0. For an object that is launched or thrown, an extra term v0t must be added to the model to account for the object’s initial vertical (upward) velocity v0 (in feet per second). Recall that h is the height (in feet), t is the time in motion (in seconds), and h0 is the initial height (in feet).h = -16t2 + h0Object is dropped
h = -16t2 + v0t + h0Object is launched
The value of v0can be ______, ______, or ______, depending on whether the object is launched upward, downward, or parallel to the ground.
Ex. 5: A basketball player passes the ball to a teammate. The ball leaves the player’s hand 5 feet above the ground and has an initial vertical velocity of 55 feet per second. The teammate catches the ball when it returns to a height of 5 feet. How long was the ball in the air?
4-10Write Quadratic Functions and Models Name:______
Objective: To Write quadratic functions and models..
Algebra 2 Standard 8.0
*In 4.1 and 4.2 sections, you learned how to graph quadratic functions. In this section, you will write quadratic functions given information about their graphs.
Example 1:Write a quadratic function in vertex form
Write a quadratic equation for the parabola shown.
y a(x h)2 k
Example 2:Write a quadratic function in intercept form
Write a quadratic equation for the parabola shown.
y a(x p)(x q)
You Try:Write a quadratic function whose graph has the given characteristics.
- x-intercepts: 2 and 1 b. vertex: (2,1)
point on graph: (1, 4) point on graph: (0, 4)
Example 3 : Write a quadratic function in standard form
Write a quadratic function in standard form for the parabola that passes through the points
(2, 6), (0, 6) and (2, 2).
Example 4: Solve a multi-step problem
The table shows the height of a baseball hit, with x representing the time (in seconds) and y representing the baseball’s height (in feet). Use a graphing calculator to find the best-fitting model for the data.
Time (x) / 0 / 2 / 4 / 6 / 8Height (y) / 3 / 28 / 40 / 37 / 26
You Try:
- Use a graphing calculator to find the best-fitting model for the data in the table.
Time (x) / 0 / 2 / 4 / 6 / 8
Height (y) / 4 / 23 / 30 / 25 / 7
- Write a quadratic function in standard form for the parabola that passes through
(1, 5), (2, 1) and (3, 1).
Alg 2 Adv Ch 4B Notes Page 1