Using the quadratic equation x2 - 6x + 8 = 0, perform the following tasks:?

Question

1) Using the quadratic equation x2 - 6x + 8 = 0, perform the following tasks:
a) Solve by factoring.







b) Solve by using the quadratic formula.



2) For the function y = x2 - 6x + 8, perform the following tasks:
a) Put the function in the form y = a(x - h)2 + k.


b) What is the equation for the line of symmetry for the graph of this function?





c) Graph the function using the equation in part a. Explain why it is not necessary to plot points to graph when using y = a (x - h) 2 + k.




Explanation of graphing.


d) In your own words, describe how this graph compares to the graph of y = x2?




3) Suppose a baseball is shot up from the ground straight up with an initial velocity of 32 feet per second. A function can be created by expressing distance above the ground, s, as a function of time, t. This function is s = -16t2 + v0t + s0
• 16 represents ½g, the gravitational pull due to gravity (measured in feet per second 2).
• v0 is the initial velocity (how hard do you throw the object, measured in feet per second).
• s0 is the initial distance above ground (in feet). If you are standing on the ground, then s0 = 0.
a) What is the function that describes this problem?




b) The ball will be how high above the ground after 1 second?




c) How long will it take to hit the ground?




d) What is the maximum height of the ball?




4) Amanda has 400 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 times width). She wants to maximize the area of her patio (area of a rectangle is length times width). What should the dimensions of the patio be, and show how the maximum area of the patio is calculated from the algebraic equation. Use the Standard Form to find the maximum area.

Answer 1

1. x^2 - 6x + 8 = 0

factor:

x^2 -4x -2x + 8 = 0

(x^2 -4x) + (-2x +8) = 0

x(x-4) -2(x-4)=0

(x-2)(x-4)=0

x=2,4

formula:

x= {6 +- sqrt(36 - 4(8))}/2

x=(6 +- 2)/2

x=8/2 & 4/2 =4,2



2. y = x2 - 6x + 8

y = (x^2 - 6x + 9) +8 -9

y=(x - 3)^2 - 1

symmetry: x=3

Don't know why you don't need to plot points





I give up, since apparently actually answering questions is frowned upon here

Answer 2

a) Solve by factorising

x2 - 6x + 8 = 0



Break up the 8 into its factors, and then identify two which add to give -6 and multiply to give 8. You should get -2 and -4.



Therefore,



(x-2)(x-4) = 0 --- ab = 0, a=0 or b=0



x= 2 or 4.



---------------------------

b) Solve using the quadratic formula.

x= [-b±√(b2 - 4ac)] / 2a



in x2 - 6x + 8 = 0,

a = 1, b = -6 and c = 8



x = [6±√(36-32)] / 2



x = [6±2]/2



x = 2 or 4

---------------------------

2) For the function y = x2 - 6x + 8, perform the following tasks:

a) Put the function in the form y = a(x - h)2 + k.



This involves completing the square.



y = x2 - 6x + 8

y = x2 - 6x + 9 - 1

y = (x-3)2-1



----------------------------

b) What is the equation for the line of symmetry for the graph of this function?



From part (a), the equation for the line of symmetry is x=h, therefore x = 3



----------------------------

c) [Graph]



It is not necessary to plot points, as you can find the equation of the line of symmetry, the vertex or turning point and intercepts very easily.



----------------------------

d) The graph has the same shape, but is shifted 1 unit down, and 3 units to the right.



----------------------------

3) You're given the general equation and all the variables, so just pop them in and you get:



s = -16t2 + 32t



---------------------------

b) Sub in t=1,



s = -16(1)2+32(1)

= 16ft.



---------------------------

c) When it hits the ground, the s = 0.



0 = -16t2 + 32t

0 = t2 - 2t

0 = t(t-2)

t = 0 or 2 seconds.



Discarding the 0 solution, we get



t = 2 seconds.



----------------------------

d) the maximum height is halfway between launch and hitting the ground, at t = 1 second.



You can use the answer from b,



16ft.



-----------------------------------

4) The equation for the perimeter is:

2(l+w) = 400ft.

l + w = 200ft.

l = 200 - w.



The equation for the area is A = l x w.



sub in (200-w) for l,



A = w(200-w)

A = - w2 + 200w



We have to find the turning (maximum) point.



The equation for the line of symmetry is -b/2a = -200/-2 = 100

Therefore at the max, w = 100ft and l = 200-100 = 100ft.



The area here is 10,000 ft2.

Answer 3

What does the word factor mean? If you do not know the definitions of math words, solving questions will not make much sense. When two integers are multiplied together, the answer is called a product. The integers that were multiplied together are called the factors of the product. For example: 5 x 6 = 30. Numbers 5 and 6 are the FACTORS and 30 is the PRODUCT. When you look for factors, you are looking for the numbers or terms that when multiplied will give you the product. We Have: x^2 - 6x + 8 = 0...ORIGINAL QUESTION Follow the steps: (1) Ask yourself: What two numbers when multiplied together will produce 8 BUT when added will produce the middle number, which in your question is -6? See it? How about -4 times -2? Yes, NEGATIVE 4 times NEGATIVE 2 = 8 but when added = -6. Is this clear? (2) WE FACTOR the trinomial. So, the trinomial x^2 - 6x + 8 becomes (x - 4) (x - 2). NOTE: (x - 4) is a FACTOR and (x - 2) is the other FACTOR. (3) Set EACH FACTOR to = 0 and solve for x. x - 4 = 0 x = 4. Now, the other factor. x - 2 = 0 x = 2 Final answer: x = 4 and x = 2. Do you know what that means? This means that if you REPLACE x in your original question with 4 or 2, you will get the SAME answer on BOTH sides. In other words, 4 and 2 are the ROOTS or SOLUTION to your ORIGINAL question. Is this clear? Again, final answer: x = 2 and x = 4. Guido

Answer 4

a) Factorization

sum=-6x and product=8

numbers are -2x and -4x

x2-4x-2x+8=0

(x2-4x)-(2x-8)=0

x(x-4)-2(x-4)=0

x-2=0 therefore x=2 or x-4=0 therefore x=4.

b)x= -b * +or -root of b2-4ac everything divided by2a

where a=1 b=-6 and c=8

x= 6 + or -root (36-32) result divide by 2

x= 6+ or -(2)/-2

x= 3+ or - 1 therefore

x=2 or x= 4

c)

Answer 5

Well, it's not so much the answering of questions that's frowned upon, as the answering of questions for people who have very clearly just pasted their entire homework assignment into their question.