11.1 Solving Quadratic Equations by the Square Root Property

• Square Root Property of Equations: If k is a positive number and if a2 = k, then

and the solution set is:

k-aka or

}, { k-k

11.1 Solving Quadratic Equations by the Square Root Property

• Example:

5

32or

5

32

325or 325

325or 325

325 2

xx

xx

xx

x

11.2 Solving Quadratic Equations by Completing the Square

• Example of completing the square:

2323

2)3(02)3(

square) the(complete 0296

factored becannot 076

22

2

2

xx

xx

xx

xx

11.2 Solving Quadratic Equations by Completing the Square

• Completing the Square (ax2 + bx + c = 0):1. Divide by a on both sides

(lead coefficient = 1)

2. Put variables on one side, constants on the other.

3. Complete the square (take ½ of x coefficient and square it – add this number to both sides)

4. Solve by applying the square root property

11.2 Solving Quadratic Equations by Completing the Square

• Review:

• x4 + y4 – can be factored by completing the square

))((

))((

(prime)

))((

2233

2233

22

22

yxyxyxyx

yxyxyxyx

yx

yxyxyx

))()(())(( 22222244 yxyxyxyxyxyx

11.2 Solving Quadratic Equations by Completing the Square

• Example:

Complete the square:

Factor the difference of two squares:

222244 yxyx

2222

22222222

2

22

xyyx

yxyyxx

xyyxxyyx 22 2222

11.3 Solving Quadratic Equations by the Quadratic Formula

• Solving ax2 + bx + c = 0:

Dividing by a:

Subtract c/a:

Completing the square by adding b2/4a2:

02 ac

ab xx

ac

ab xx 2

2

2

2

2

44

2

ab

ac

ab

ab xx

11.3 Solving Quadratic Equations by the Quadratic Formula

• Solving ax2 + bx + c = 0 (continued): Write as a square:

Use square root property:

Quadratic formula:

2

2

4442

2 4

42

2

2

a

acbx

ab

aac

ab

a

acb

a

bx

2

4

2

2

a

acbbx

2

42

11.3 Solving Quadratic Equations by the Quadratic Formula

• Quadratic Formula:

is called the discriminant.If the discriminant is positive, the solutions are realIf the discriminant is negative, the solutions are imaginary

a

acbbx

2

42

acb 42

11.3 Solving Quadratic Equations by the Quadratic Formula

• Example:

2,32

1

2

5

2

24255

)1(2

)6)(1(4)5()5(

6c -5,b 1,a 065

2

2

xx

x

xx

11.3 Solving Quadratic Equations by the Quadratic Formula

• Complex Numbers and the Quadratic FormulaSolve x2 – 2x + 2 = 0

i

ii

x

12

22

2

42

2

42

)1(2

)2)(1(4)2()2( 2

11.4 Equations Quadratic in Form

Method Advantages Disadvantages

Factoring Fastest method Not always factorable

Square root property

Not always this form

Completing the square

Can always be used

Requires a lot of steps

Quadratic Formula

Can always be used

Slower than factoring

bax 2)( :form

11.4 Equations Quadratic in Form

• Sometimes a radical equation leads to a quadratic equation after squaring both sides

• An equation is said to be in “quadratic form” if it can be written as a[f(x)]2 + b[f(x)] + c = 0

Solve it by letting u = f(x); solve for u; then use your answers for u to solve for x

11.4 Equations Quadratic in Form

• Example:

Let u = x2

03434 22224 xxxx

1,31,3

1,30)1)(3(

034034

22

2222

xxxx

uuuu

uuxx

11.5 Formulas and Applications

• Example (solving for a variable involving a square root)

4by sidesboth divide 4

by sidesboth multiply 4

sidesboth square 4

Afor 4

:Solve

2

2

2

Ad

Ad

Ad

Ad

11.5 Formulas and Applications

• Example:

4

8 and

4

8 so

4

8

formula) quadratic( )2(2

))(2(4

sideright on zeroget 20

for t solvekt 2

22

2

2

2

2

sk-kt

sk-kt

sk-k

sk-kt

sktt

ts

11.6 Graphs of Quadratic Functions

• A quadratic function is a function that can be written in the form:f(x) = ax2 + bx + c

• The graph of a quadratic function is a parabola. The vertex is the lowest point (or highest point if the parabola is inverted

11.6 Graphs of Quadratic Functions

• Vertical Shifts:

The parabola is shifted upward by k units or downward if k < 0. The vertex is (0, k)

• Horizontal shifts:

The parabola is shifted h units to the right if h > 0 or to the left if h < 0. The vertex is at (h, 0)

kxxf 2)(

2)( hxxf

11.6 Graphs of Quadratic Functions

• Horizontal and Vertical shifts:

The parabola is shifted upward by k units or downward if k < 0. The parabola is shifted h units to the right if h > 0 or to the left if h < 0 The vertex is (h, k)

khxxf 2)(

11.6 Graphs of Quadratic Functions

• Graphing:

1. The vertex is (h, k).

2. If a > 0, the parabola opens upward.If a < 0, the parabola opens downward (flipped).

3. The graph is wider (flattened) if

The graph is narrower (stretched) if

khxaxf 2)(

10 a

1a

11.6 Graphs of Quadratic Functions Inverted Parabola with Vertex (h, k)

Vertex = (h, k)

khxxf 2)(

11.7 More About Parabolas; Applications

• Vertex Formula:The graph of f(x) = ax2 + bx + c has vertex

a

bf

a

b

2,

2

11.7 More About Parabolas; Applications

• Graphing a Quadratic Function:

1. Find the y-intercept (evaluate f(0))

2. Find the x-intercepts (by solving f(x) = 0)

3. Find the vertex (by using the formula or by completing the square)

4. Complete the graph (plot additional points as needed)

11.7 More About Parabolas; Applications

• Graph of a horizontal (sideways) parabola:The graph of x = ay2 + by + c or x = a(y - k)2 + his a parabola with vertex at (h, k) and the horizontal line y = k as axis. The graph opens to the right if a > 0 or to the left if a < 0.

11.7 More About ParabolasHorizontal Parabola with Vertex (h, k)

Vertex = (h, k)

hkyx 2