Chapter 5 – Factoring and Solving Quadratics

16 Days

Factoring QuadraticsTwo Days

Review - Use FOIL and the Distributive Property to multiply polynomials.

Warm up -Multiplying Polynomials

)5)(12( xx

)5)(3( xx

Writing a polynomial as a product of its factors.

Essentially undoing the multiplication. Purposes of factoring:

◦ Simplifying◦ Rewriting◦ Solving

Factoring Polynomials

162x 24102 xx

GCF

Types of Factoring

Grouping

Types of Factoring

Difference of Squares

Types of Factoring

Difference of Squares Formula

Types of Factoring

))((22 bababa

Perfect Square Trinomials

Types of Factoring

962 xx 22 44 yxyx

Perfect Square Trinomial Formulas

Types of Factoring

222 )(2 bababa

222 )(2 bababa

Trinomials (a=1)

Types of Factoring

Trinomials (a≠1)

Types of Factoring

352 2 xx 26 2 xx

Sum and Difference of Cubes Formulas

Types of Factoring

)2)(( 2233 yxyxyxyx

)2)(( 2233 yxyxyxyx

Sum and Difference of Cubes

Types of Factoring

pg 263 (# 2-30 even)

Homework

Factoring Practice

492

1811

183

16

67

2

2

2

2

2

xx

xx

xx

x

xx

More Factoring Practice

2516

656

31310

462

9

2

2

2

2

3

x

xx

xx

xx

xx

Solving Quadratics by Factoring

Two Days

Question: How can we multiply two or more numbers together and get a product that equals zero?

For any real numbers a and b, if ab=0, then either a=0, b=0, or both.

Zero Product Property

In this first example, the equation is already factored and is set equal to zero. To solve, simply set the individual factors equal to zero.

x 3 2x 1 0

x 3 0 or 2x 1 0

x 3 or 2x 1

x 3 or x 1

2

The solutions are -3 and 1/2.

In this example, you must first factor the equation. Notice the familiar pattern. After factoring, set the individual factors equal to zero.

9x2 4 0

3x 2 3x 2 03x 2 0 or 3x 2 0

3x 2 or 3x 22 2

or3 3

x x

Factor using “difference of two squares.”

In the next example, you must set the equation equal to zero before factoring. Then set the individual factors equal to zero and solve.

x2 6x 27 0x 9 x 3 0

x 9 0 or x 3 0

9 or 3x x

x2 27 6x

This one uses a different technique than the previous ones. Really, this is something you should consider at the beginning of every factoring problem. See if you can solve it.

2x x 4 0

2x 0 or x 4 0

0 or 4x x

2x2 8x 0Did you take out GCF?

Now, try several problems. Write these on your own paper, showing all steps

carefully. 1. 3y 5 2y 7 0

2. x2 x 12

3. d2 5d 0

4. 4c2 25

5. 18u2 1 3u

Here are the answers. For help, click on the

numbers.

1. 2. 3. 4. 5.

If all are correct, you’re finished!

y5 3 or y 7 2

x 4 or x3

d 0 or d 5

c 5 2 or c 5 2

u 1 6 or u 1 3

3y 5 2y 7 0

3y 5 0 or 2y 7 0

3y 5 or 2y 7

y5 3 or y 7 2

Back to questions

x2 x 12

x2 x 12 0

x 4 x 3 0

x 4 0 or x 3 0

x 4 or x3 Back to questions

d2 5d 0

d d 5 0

d 0 or d 5 0

d 0 or d 5

Back to questions

4c2 25

4c2 25 0

2c 5 2c 5 0

2c 5 0 or 2c 5 0

2c 5 or 2c5

c 5 2 or c 5 2

Back to questions

18u2 3u 1

18u2 3u 1 0

6u 1 3u 1 0

6u 1 0 or 3u 1 0

6u 1or 3u 1

u 1 6 or u 1 3Back to questions

Solving Quadratics Using Other Methods

One Day

Solve by taking Square Root

Complex NumbersTwo Days

Solve the following quadratic:

Warm-up

042 x

The Imaginary Unit (i) has the following properties.

The imaginary number i is defined as the number whose square is -1. That is:

Imaginary Numbers are of the form a + bi where b ≠ 0.

Complex Numbers are of the form a + bi where a and b are Real Numbers.

Imaginary Numbers

We can add and subtract imaginary numbers similar to how we add and subtract terms with variables. Think “like terms.”

Addition of Imaginary Numbers

Similarly, we can multiply imaginary numbers following the same exponent rules we use for variables.

Multiplying Imaginary Numbers

The absolute value of a complex number is the distance the number lies from the origin in the complex plane.

Think Pythagorean Thm..

Absolute Value of Complex Numbers

22 babia

Larger powers of i can be simplified by dividing the power by 4 and using the remainder to determine the appropriate value.

Powers of i

Solve:

Finding Complex Solutions

02005 2 x

Pg 278 (# 1-45 odd)

Homework

If z = a + bi is an imaginary number, the its conjugate is z = a – bi.

Complex Conjugates can be used to eliminate imaginary numbers from the denominators of fractions. This is very similar to how we rationalize denominators.

Complex Conjugates

Eliminate the Imaginary numbers from the denominator in the following example.

Using Complex Conjugates

Practice 5-6 WS (even)

Homework

Completing the Square and the Quadratic Formula

Two Days

Perfect Square Trinomials

Examplesx2 + 6x + 9x2 - 10x + 25x2 + 12x + 36

Creating a Perfect Square Trinomial

In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____

Find the constant term by squaring half the coefficient of the linear term.

(14/2)2

X2 + 14x + 49

Solve for x..

02 cbxax

a

acbbx

2

4 :Formula Quardatic

2

Solve the following using the quadratic formula:

Using the Quadratic Formula

0562 xx

0932 2 xx

What do we notice about these two problems? How else could we Have solved these quadratics?

Solve the following quadratics:

Using the Quadratic Formula

05103 2 xx

01043 2 xx

pg 293 (# 1-29 odd)

Homework

Quadratic Formula and the Discriminant

1 Day

THE QUADRATIC FORMULA

1. When you solve using completing the square on the general formula you get:

2. This is the quadratic formula!3. Just identify a, b, and c then substitute

into the formula.

2 4

2

b b acx

a

2 0ax bx c

WHY USE THE QUADRATIC FORMULA?

The quadratic formula allows you to

solve ANY quadratic equation, even

if you cannot factor it.

An important piece of the quadratic

formula is what’s under the radical:

b2 – 4ac

This piece is called the discriminant.

WHY IS THE DISCRIMINANT IMPORTANT?

The discriminant tells you the number and types of

answers

(roots) you will get. The discriminant can be +, –, or 0

which actually tells you a lot! Since the discriminant is

under a radical, think about what it means if you have a

positive or negative number or 0 under the radical.

WHAT THE DISCRIMINANT TELLS YOU!

Value of the Discriminant

Nature of the Solutions

Negative 2 imaginary solutions

Zero 1 Real Solution

Positive – perfect square

2 Reals- Rational

Positive – non-perfect square

2 Reals- Irrational

Use the discriminant to determine the type and number of roots.

0542 xx

0282 2 xx

0526 2 xx

036122 xx

Pg 293 (#31-39 odd) Practice 5-8 WS (#2-26 even)

Quiz 5.8 on 11/18!!

Homework

5.1 Modeling with Quadratics

1 Day

The standard form of a quadratic is

Standard Form of a Quadratic

erm.constant t theis

and m,linear ter theis

term,quadratic theis

: where2

2

c

bx

ax

cbxaxy

The graph of a quadratic function is called a parabola.

The axis of symmetry is the vertical line that divides the parabola into two identical halves and is written x=a.

The vertex (a,b) of the parabola is the point at which the parabola intersects the axis of symmetry and is also a maximum or minimum point of the function.

Parabolas

Given 3 points on the function we can determine the equation of the quadratic.

Writing Equations of Quadratics

function. on the points are 10)(3, and 3),(2, (1,0),

: where Find 2

cbxaxy

pg. 241 (#1-12 all, 21)

Homework

5.2 Graphing Parabola4 Days

5.3 Transforming Parabolas

3 Days

Standard form of a Quadratic:

Vertex form of a Quadratic:

Quadratic Functions

0 ; 2 acbxaxy

),( :Vertex ; )( 2 khkhxay

Competing the Square

463 2 xxy

4) 2(3 2 xxy222 )1(34))1( 2(3 xxy

3a (-1,-7),:Vertex 7)1(3 2 xy