1. Chapter 7 Quadratic Equations
TMAT 103
Chapter 7
Quadratic Equations

2. §7.1 Solving Quadratic Equations by Factoring
TMAT 103
§7.1
Solving Quadratic Equations by Factoring

3. §7.1 – Solving Quadratic Equations by Factoring
Quadratic Equation – general form:
Key principle – Zero Factor Property:
If ab = 0, then either a = 0, b = 0, or both

4. §7.1 – Solving Quadratic Equations by Factoring
Solving a Quadratic Equation by Factoring (b  0)
If necessary, write the equation in the form ax2 + bx + c = 0
Factor the nonzero side of the equation
Using the preceding problem, set each factor that contains a variable equal to zero
Solve each resulting linear equation
Check

5. §7.1 – Solving Quadratic Equations by Factoring
Examples – Solve the following by factoring
x2 – 6x + 8 = 0
2x2 + 9x = 5
x – 2x2 = 0

6. §7.1 – Solving Quadratic Equations by Factoring
Solving a Quadratic Equation by Factoring (b = 0)
If necessary, write the equation in the form ax2 = c
Divide each side by a
Take the square root of each side
Simplify the result, if possible

7. §7.1 – Solving Quadratic Equations by Factoring
Examples – Solve the following by factoring
4x2 = 9
16 – x2 = 0

8. §7.2 Solving Quadratic Equations by Completing the Square
TMAT 103
§7.2
Solving Quadratic Equations by Completing the Square

9. §7.2 – Solving Quad Equations by Completing the Square
Solving a Quadratic Equation by Completing the Square
The coefficient of the second-degree term must equal (positive) 1. If not, divide each side of the equation by its coefficient
Write an equivalent equation in the form x2 + px = q.
Add the square of ½ of the coefficient of the linear term to each side; that is, (½p)2
The left side is now a perfect square trinomial. Rewrite the left side as a square
Take the square root of each side
Solve for x and simplify, if possible
Check

10. §7.2 – Solving Quad Equations by Completing the Square
Examples – Solve the following by completing the square
x2 – 6x + 8 = 0
2x2 + 9x = 5
x – 2x2 = 0

11. §7.3 The Quadratic Formula
TMAT 103
§7.3 The Quadratic Formula

12. §7.3 The Quadratic Formula
The general quadratic equation can now be solved by completing the square
This will generate a formula that can be used to solve any quadratic equation
x will be written in terms of a, b, and c

13. §7.3 The Quadratic Formula
Solving a Quadratic Equation using the Quadratic Formula
If necessary, write the equation in the form ax2 + bx + c = 0
Substitute a, b, and c into the quadratic formula
Solve for x
Check

14. §7.3 The Quadratic Formula
Examples – Solve the following by using the quadratic formula
x2 – 6x + 8 = 0
2x2 + 9x = 5
x – 2x2 = 0

15. §7.3 The Quadratic Formula
Consider the quadratic formula
The discriminant provides insight into the nature of the solutions
discriminant

16. §7.3 The Quadratic Formula
Discriminant
If b2 – 4ac > 0, there are 2 real solutions
If b2 – 4ac is also a perfect square they are both rational
If b2 – 4ac is not a perfect square, they are both irrational
If b2 – 4ac = 0, there is only one rational solution
If b2 – 4ac < 0, there are two imaginary solutions
Chapter 14

17. §7.3 The Quadratic Formula
Examples – How many and what types of solutions do each of the following have?
x2 – 2x + 17 = 0
x2 – x – 2 = 0
x2 + 6x + 9 = 0
2x2 + 2x + 14 = 0

18. TMAT 103
§7.4
Applications

19. §7.4 Applications Examples
The work done in Joules in a circuit varies with time in milliseconds according to the formula w = 8t2 – 12t Find t in ms when w = 16J.
A rectangular sheet of metal 24 inches wide is formed into a rectangular trough with an open top and no ends. If the cross-sectional area is 70 in2, find the depth of the trough.

20. §14.1 Complex Numbers in Rectangular Form
TMAT 103
§14.1
Complex Numbers in Rectangular Form

21. §14.1 – Complex Numbers in Rectangular Form
Imaginary Unit
In mathematics, i is used
In technical math, i denotes current, so j is used to denote an imaginary number
Rectangular Form of a Complex Number
a is the real component, and bj is the imaginary component

22. §14.1 – Complex Numbers in Rectangular Form
Examples – Express in terms of j and simplify

23. §14.1 – Complex Numbers in Rectangular Form
Powers of j
j = j
j2 = –1
j3 = –j
j4 = 1
j5 = j
j6 = –1
j7 = –j
j8 = 1
… Process continues
Powers of j evenly divisible by four are equal to 1

24. §14.1 – Complex Numbers in Rectangular Form
Examples – Express in terms of j and simplify

25. §14.1 – Complex Numbers in Rectangular Form
Additional Information
Complex numbers are not ordered
“Greater than” and “Less than” do not make sense
Conjugate
The conjugate of (a + bj) is (a – bj)

26. §14.1 – Complex Numbers in Rectangular Form
Addition and subtraction
Complex numbers can be added and subtracted as if they were 2 ordinary binomials
(a + bj) + (c + dj) = (a + c) + (b + d)j
(a + bj) – (c + dj) = (a – c) + (b – d)j

27. §14.1 – Complex Numbers in Rectangular Form
Examples – Perform the indicated operation
(1 – 2j) + (3 – 5j)
(–3 + 13j) – (4 – 7j)
(½ – 11j) – (½ – 4j)

28. §14.1 – Complex Numbers in Rectangular Form
Multiplication
Complex numbers can be multiplied as if they were 2 ordinary binomials
(a + bj)(c + dj) = (ac – bd) + (ad + bc)j

29. §14.1 – Complex Numbers in Rectangular Form
Examples – Multiply
(1 – 2j)(3 – 5j)
(–3 + 13j)(4 – 7j)
(½ – 11j)(½ – 4j)

30. §14.1 – Complex Numbers in Rectangular Form
Division
Complex numbers can be divided by multiplying numerator and denominator by the conjugate of the denominator

31. §14.1 – Complex Numbers in Rectangular Form
Examples – Divide

32. §14.1 – Complex Numbers in Rectangular Form
Solving quadratic equations with a negative discriminant
2 complex solutions
Always occur in conjugate pairs
Use quadratic formula, or other techniques

33. §14.1 – Complex Numbers in Rectangular Form
Examples – Solve using the quadratic formula
x2 + x + 1 = 0
x2 + 9 = 0

34. §14.1 – Complex Numbers in Rectangular Form
Adding complex numbers graphically

35. §14.1 – Complex Numbers in Rectangular Form
Subtracting complex numbers graphically