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TMAT 103 Chapter 7 Quadratic Equations

TMAT 103 Chapter 7 Quadratic Equations. TMAT 103 §7.1 Solving Quadratic Equations by Factoring

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TMAT 103

Chapter 7

Quadratic Equations

TMAT 103

§7.1

Solving Quadratic Equations by Factoring

§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

TMAT 103

§7.2

Solving Quadratic Equations by Completing the Square

§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

TMAT 103

§7.3The Quadratic Formula

§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

TMAT 103

§7.4Applications

§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.

TMAT 103

§14.1

Complex Numbers in Rectangular Form

§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

• Examples – Express in terms of j and simplify

48

81

§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

• Examples – Divide

j

jj

j

j

3

442

32

76

§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

§14.1 – Complex Numbers in Rectangular Form

• Adding complex numbers graphically

§14.1 – Complex Numbers in Rectangular Form

• Subtracting complex numbers graphically