Math I

UNIT QUESTION: What is a quadratic function?Standard: MM2A3, MM2A4

Today’s Question:How do you solve quadratic equations that can’t be factored?Standard: MM2A4.b.

DAY ONE

3.6 - Solving a Quadratic Equation by Square Roots

Let’s look at some examples where x2 is already by

itself.

Examples. Solve the equation. Write the solutions as integers if possible. Otherwise, write them as radical expressions.

5.24.1 22 nx

42 x 52 n

5n2x

10.425.311.2811.

:You try2222 xcyx

Let’s look at some examples where

x2 is NOT by itself.

0483 2 xWe must solve to get x2 by

itself 1st!

483 2 x

162 x

4x

96322 xWe must solve to get x2 by

itself 1st!

642 x

8x

You try!

0722 01 22 xx

Falling object model

When an object is dropped, the speed with which it falls continues to increase. Ignoring air resistance, its height h can be approximated by the falling object model:

sth 216

An engineering student is a contestant in an egg dropping contest. The goal is to create a container for an egg so it can be dropped from a height of 32 feet without breaking (s = starting height). Find the time (t) it will take for the egg to hit the ground (height = ? ). Disregard air resistance.

SOLUTION: The starting height is 32 feet.

Now, substitute 0 for h and solve.32160 2 t21632 t Subtract 32 from both sides

Divide both sides by –16 22 tTake the square root of both sides

4.12 t So, the answer is 1.4 seconds. It is only the positive of the square root b/c you can’t have negative seconds!!!!!

sth 216

DAY TWO

The following quadratics are in standard form, ax2 + bx + c

Evaluate b2 – 4ac for each of the following

1

0

-15

1)x2 – 3x + 2

2)x2 – 4x + 4

3)2x2 – 3x + 3

There is a way to tell how many roots an equation will have.

It’s called finding the discriminant.

The discriminant is a small part of the quadratic formula.

b ac2 4

b ac2 4If the answer is POSITIVE, then you will have 2 roots.

If the answer is ZERO, then you will have 1 root.

If the answer is NEGATIVE, then you will have no roots.

Our text book says solutions instead of roots (same thing).

Determine the number of roots.

Example: 1

0432 xx

b ac2 4 ( ) ( )( )3 4 1 42

9 16 7

Determine the number of roots.

Example: 2

0442 xx

b ac2 4 ( ) ( )( )4 4 1 42

16 160

Determine the number of roots.

Example: 3

0452 xx

b ac2 4 ( ) ( )( )5 4 1 42

25 169

Find the number of x-intercepts.Example: 4

342 xxy

b ac2 4 ( ) ( )( )4 4 1 32

16 124

Find the number of x-intercepts.Example: 5

322 xxy

b ac2 4 )3)(1(4)2( 2 124

8

Solve for x:

2 25 0x

2 9 14 0x x

Quadratic Formula: gives the solution of in terms of the coefficients a, b & c.

2 0ax bx c

a

acbbx

2

42

a

The solutions of the quadratic equation are

a

a

c

c

bb

b

Ex: 1 Solve x2+ 9x +14 = 0

a

acbbx

2

42

a =

b =

c =

12

141499 2

1st

12

1414819

2nd

2

56819

2

259

2

59

2

59

2

59 and

2

14

2

4 and 7 2 and

2 4 3 0x x

2

2

2

1.)2 10 0

2.) 3 4 0

3.)2 3 8

x x

x x

x x

-5/2 & 2

-1 & 4

2.89 & -1.39

A rocket is shot upward with an initial velocity of 125 feet per second from a platform 3 feet above the ground. Use the model to find how long it will take the rocket to hit the ground.

216h t vt s

3 ft

t = 7.83 seconds

HOMEWORK