Name __________________________________________________________________ Period ____________

Algebra 3-4 Unit 2 Part 2 Quadratics

2.7 I can simplify an expression with i.

2.8 I can add, subtract, and multiply with i.

2.9 I can solve quadratic equations with imaginary solutions.

2.10 I can graph quadratics and identify key features of the graph.

2.11 I can solve systems of equations.

My goal for this unit: _____________________________________________________ ______________________________________________________________________ What I need to do to reach my goal: ________________________________________ ______________________________________________________________________ ______________________________________________________________________

Name ___________________________________________________________________ Period ___________

Algebra 3-4 2.7 Use Your Imagination

Taking the square root of a negative does not provide a real solution.

It does, however, provide an imaginary solution.

The value i is an imaginary number equal to the

square root of −1.

i = -1

A complex number has both a real part and an imaginary part.

a + bi

5 + 3i

Similar to combining like terms, real parts can be combined only with real parts and imaginary

parts only with imaginary parts.

(6 + 2i) + (7 − 5i)

6 + 7 + 2i − 5i

13 − 3i

The table below shows the value of i when raised to a power.

i1 i2 i3 i4

i = i -1e j2

= −1 (i2)(i1) = (−1)(i) = −i (i2)(i2) = (−1)(−1) = 1

For values of i raised to higher powers, this same pattern repeats:

i5 = (i4)(i1)

= 1i

= i

i6 = (i4)(i2)

= 1(−1)

= −1

i7 = (i4)(i3)

= 1(−i)

= −i

i8 = (i4)(i4)

= 1(1)

= 1

When given a large power of i, rewrite using a power that is a multiple of 4.

i26 = (i24)(i2) = (i4)6(i2) = (1)6(−1) = −1

To simplifying radicals with negative values, convert into imaginary numbers first.

-7 -7 -1 -7= = = =i i i7 7 49 72 c hc h

real

part

imaginary

part

They are called “imaginary” because

people could not imagine that they

could exist. But they are no less real

than “real” numbers since both are

inventions of people.

Directions: Simplify each of the following expressions.

1. 25− 2. 100−

3. i98 4. i53

5. i44 6. i75

7. (i23)2 8. (i25)(i24)

9. -2 -18 10. -3 -4 -5

11. (i3)5(i2) 12. -3 -21

13. -8 -12 -16 14. i103

Name ___________________________________________________________________ Period ___________

Algebra 3-4 2.8 Operations on Imaginary Numbers

Add or subtract complex numbers by combining their real and imaginary parts separately. Add: Subtract: Distribute when necessary: (3 + 5i) + (2 + 4i) (3 + 5i) − (2 + 4i) 3(2 + 6i) − 4(2 − 3i) 3 + 5i + 2 + 4i 3 + 5i − 2 − 4i 6 + 18i − 8 + 12i 5 + 9i 1 + i −2 + 30i Apply FOIL when necessary: (4 − 9i)(2 + 5i) (2 + i)(6 − 2i) 8 + 20i − 18i − 45i2 12 − 4i + 6i − 2i2 8 + 2i − 45(-1) 12 + 2i − 2(−1)

8 + 2i + 45 12 + 2i + 2 53 + 2i 14 + 2i Directions: Simplify each expression. Write all answers in the form a + bi. 1. (12 − 5i) + (-9 − 2i) 2. (2 − 5i) − (-3 + 7i)

3. (10 − 7i) − 2(3 + 9i) 4. (7 − 3i) − (-2 − 4i)

5. 3(5 − 2i) + 4(2 + i) 6. -2(6 − 3i) + 3(8 + 3i)

7. 5(3 + 7i) + 12 − 9i 8. 3(2 − 9i) − 4(2 + i)

i2 = -1

i2 = -1

9. (9 − 3i)(-3 + 5i) 10. i 79

11. (5 + 3i) − (9 − 2i) 12. (7 − 3i)(8 + 2i)

13. (2 − 4i)(5 + i) 14. (4 − 3i)2

15. 3i(6 − 2i) + 5(2 + 3i) 16. 2i(1 − i) + 3(3 − 4i)

17. Simplify (i 47)3 18. -5 -5

19. 4(2 + 8i) + (3 − 5i)2 20. 4(3 − 7i) + 5(2 + i)

21. (5 − 2i)(3 + 8i) 22. (5 − 2i)2

Name ___________________________________________________________________ Period ___________

Algebra 3-4 2.9 Solve with Imaginary Solutions

Solve each equation.

1. 5x2 − 3x + 1 = 0 2. 2x2 − 2x + 5 = 0

3. 3x2 + x + 2 = 0 4. x2 + x + 5 = 0

5. 5x2 − x + 2 = 0 6. 4x2 − x + 3 = 0

7. 3x2 − 4x − 1 = 0 8. x2 + 10 = 5x

9. x2 − 4x = −5 10. x2 + 6x + 13 = 0

11. x2 + 4x + 2 = 0 12. 16x2 − 11x + 9 = 13x

13. x2 − 7x − 3 = 0 14. x2 + 8 = 5x

Name ___________________________________________________________________ Period ___________

Algebra 3-4 2.10 Graphing Quadratics

Graph each given equation and then fill in the blanks.

1. y = x2 − 2x − 15 Vertex: x-intercept(s): y-intercept: domain: range: axis of symmetry: factored form: vertex form:

2. y = x2 − 4

Vertex: x-intercept(s): y-intercept: domain: range: axis of symmetry: factored form: vertex form:

y

x

y

x

3. y = 3x2 − 6x + 5

Vertex: x-intercept(s): y-intercept: domain: range: axis of symmetry:

4. y = −x2 + 8x Vertex: x-intercept(s): y-intercept: domain: range: axis of symmetry: factored form:

y

x

y

x

5. y = x2 + 4x + 4 Vertex: x-intercept(s): y-intercept: domain: range: axis of symmetry: factored form: vertex form:

6. y = 2x2 − 12x + 3 Vertex: x-intercept(s): y-intercept: domain: range: axis of symmetry:

y

x

y

x

7. y = −x2 + 10x − 1

Vertex: x-intercept(s): y-intercept: domain: range: axis of symmetry:

8. y = 6x2 + x − 15 Vertex: x-intercept(s): y-intercept: domain: range: axis of symmetry: factored form:

y

x

y

x

Name ___________________________________________________________________ Period ___________

Algebra 3-4 2.11 Solving Systems

A system of equations is two or more equations. To solve, you want to find the value(s) that work in all of the equations. Given the system of equations: y = 4x − 9 and y = x − 3

The ordered pair (2, −1) works in both equations: −1 = 4(2) − 9 and −1 = 2 − 3 Therefore, this is the solution to the system. One way to solve for this solution is using substitution: y = 4x − 9 and y = x − 3

4x − 9 = x − 3

3x − 9 = −3

3x = 6 x = 2 y = (2) − 3 = −1

(2, −−−−1) Use the method of substitution to solve for the solution to each system.

1. y = −3x + 4 y = 3x − 2

2. y = −2x + 5 y = x − 13

Now use the graphing calculator or desmos to verify the answers you just found. Solve each of the following systems using any method you want.

3. y = x 2 + 3x − 5 y = x + 3

4. y = x 2 − 4x + 6 y = x + 2

5. y = x 2 − 10x + 14 y = 7x − 16

6. y = x 2 − 24 y = x − 12

7. y = x 2 − 8x − 12 y = 4x + 8

8. y = 2x2 + 3 y = x + 2

9. y = x 2 − 9x − 18 y = x + 3

10. y = x 2 + 6x + 10 y = −2x − 6

11. y = x 2 − 6x + 10 y = 1

12. y = x 2 − 3x − 6 y = x + 6

13. y = x 2 + 6x + 3 y = 3x − 7

14. y = x 2 + 8x + 16 y = x + 6

Name _______________________________________________________________ Period _______________

Are You Ready for Unit 2 (part 2) Test?

I can simplify and solve using imaginary numbers.

1. Simplify: 81−

2. Simplify: 121− 3. Simplify: 25−

4. Simplify: (5 − 2i)(6 + 4i)

5. Simplify: (7 + 3i)(2 + 5i) 6. Simplify: (8 − 3i)(11 − 3i)

7. Simplify: (9 + 2i)2

8. Simplify: (8 − 3i)2 9. Simplify: (3 − i)2

10. Simplify: i 102

11. Simplify: i 88 12. Simplify: i 87

13. Simplify: (4 + 2i) + (9 − 3i)

14. Simplify: (6 + 3i) − (3 + 7i) 15. Simplify: (8 − 3i) − (6 − 2i)

16. Simplify: (3 − 5i) − 6i (3 − 8i)

17. Simplify: (3 − 2i) − 4i (2 + 9i) 18. Simplify:

(−2 − 3i) − 8i (2 − 5i)

19. State if real or imaginary: (4i)16

20. State if real or imaginary: (4i)17

21. State if real or imaginary: (4i)18

22. State if real or imaginary: (4i)19

23. State if real or imaginary: (3 − 3i)2

24. State if real or imaginary: (2 + 5i)2

25. State if real or imaginary:

(1 − 7i)2

26. State if real or imaginary:

(3 − 5i)(3 + 5i) 27. State if real or imaginary:

(9 + 3i)(9 − 3i)

28. State if real or imaginary:

(6 − 7i)(6 − 7i)

29. State if real or imaginary:

(2 + 9i)(2 − 9i) 30. State if real or imaginary:

(3 + 2i)(3 − 2i)

31. Solve using the quadratic formula: x

2 − 6x + 12 = 0

32. Solve using the quadratic formula: x

2 + 2x + 8 = 0

33. Solve using the quadratic formula: x

2 + 3x + 11 = 0

I can graph quadratics and identify the key features.

34. Given the quadratic function f(x) = x 2 − 8x + 6.

Write in vertex form. What is the vertex?

35. Given the quadratic function f(x) = x 2 + 4x + 2. Write in vertex form. What is the vertex?

36. y = x 2 + 2x + 2 Vertex:

x-intercept(s):

y-intercept:

domain:

range:

axis of symmetry:

factored form:

vertex form:

I can solve systems of equations.

37. Solve: y = x 2 − 8x − 12

y = 4x − 8

38. Solve: y = x 2 − 4x − 20

y = x − 5

39. Solve: y = x2 − 2x + 2

y = x − 1