§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
0 where02 acbxax
§7.1 – Solving Quadratic Equations by Factoring
• Solving a Quadratic Equation by Factoring (b 0)
1. If necessary, write the equation in the form ax2 + bx + c = 0
2. Factor the nonzero side of the equation3. Using the preceding problem, set each factor
that contains a variable equal to zero4. Solve each resulting linear equation5. Check
§7.1 – Solving Quadratic Equations by Factoring
• Examples – Solve the following by factoring
x2 – 6x + 8 = 0
2x2 + 9x = 5
x – 2x2 = 0
§7.1 – Solving Quadratic Equations by Factoring
• Solving a Quadratic Equation by Factoring (b = 0)
1. If necessary, write the equation in the form ax2 = c
2. Divide each side by a
3. Take the square root of each side
4. Simplify the result, if possible
§7.1 – Solving Quadratic Equations by Factoring
• Examples – Solve the following by factoring
4x2 = 9
16 – x2 = 0
§7.2 – Solving Quad Equations by Completing the Square
• Solving a Quadratic Equation by Completing the Square
1. The coefficient of the second-degree term must equal (positive) 1. If not, divide each side of the equation by its coefficient
2. Write an equivalent equation in the form x2 + px = q.3. Add the square of ½ of the coefficient of the linear term to
each side; that is, (½p)2
4. The left side is now a perfect square trinomial. Rewrite the left side as a square
5. Take the square root of each side6. Solve for x and simplify, if possible7. Check
§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
§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
0 where02 acbxax
§7.3 The Quadratic Formula
• Solving a Quadratic Equation using the Quadratic Formula
1. If necessary, write the equation in the form ax2 + bx + c = 0
2. Substitute a, b, and c into the quadratic formula
3. Solve for x
4. Check
a
acbbx
2
42
§7.3 The Quadratic Formula
• Examples – Solve the following by using the quadratic formula
x2 – 6x + 8 = 0
2x2 + 9x = 5
x – 2x2 = 0
§7.3 The Quadratic Formula
• Consider the quadratic formula
• The discriminant provides insight into the nature of the solutions– discriminant
a
acbbx
2
42
acb 42
§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
§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
§7.4 Applications
• Examples– The work done in Joules in a circuit varies with
time in milliseconds according to the formula w = 8t2 – 12t + 20. 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.
§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
1j
bja
§14.1 – Complex Numbers in Rectangular Form
• Powers of jj = jj2 = –1j3 = –jj4 = 1j5 = jj6 = –1j7 = –jj8 = 1… Process continues
• Powers of j evenly divisible by four are equal to 1
§14.1 – Complex Numbers in Rectangular Form
• Examples – Express in terms of j and simplify
15303
211
22
jj
j
j
§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)
§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
§14.1 – Complex Numbers in Rectangular Form
• Examples – Perform the indicated operation(1 – 2j) + (3 – 5j)
(–3 + 13j) – (4 – 7j)
(½ – 11j) – (½ – 4j)
§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
§14.1 – Complex Numbers in Rectangular Form
• Examples – Multiply
(1 – 2j)(3 – 5j)
(–3 + 13j)(4 – 7j)
(½ – 11j)(½ – 4j)
§14.1 – Complex Numbers in Rectangular Form
• Division– Complex numbers can be divided by
multiplying numerator and denominator by the conjugate of the denominator
jdc
adbc
dc
bdac
djc
bja
2222
§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
§14.1 – Complex Numbers in Rectangular Form
• Examples – Solve using the quadratic formula
x2 + x + 1 = 0
x2 + 9 = 0