Help with these math problems please?

Question

1. If 3x-y=9 and {x| -6 greater than or equal to x less than or equal to 7} what is the range of y?

2. What is the range of the quadratic function y= 2x^2 - 6x - 7 ?

3. Find the product of 3 consecutive whole numbers if the largest is n+1.

4. If P(x)= x^4 - x^3 - 5x^2 + 9x - 1 and D(x)= x^2 - 2x + 3, and P(x) is written in the form P(x) = D(x)Q(x) + R(x), what is R(x)?

5. Find the set of possible rational roots of 3x^4 - 4x^3 + 7x - 6.

6. Find all the roots of 2x^3 - 5x^2 - x + 6 = 0.

7. Find the area of the triangle formed by the x-axis, y-axis and the straight line 3x+2y-12=0.

8. An object moves 64cm in the first second, 48cm in the 2nd, 36cm in the 3rd and so on. Find the total distance it will move by the time it stops.

Please provide solutions/explanations, even if just the equations needed to solve the problem. I can take it from there. Thanks!

Answer 1

1. 3x - y = 9, and -6 <= x <= 7. Find the range of y.

To solve this, you must first solve for y, in the equation 3x - y = 9.
After all, the definition of range is the values y can be.

3x - y = 9
-y = -3x + 9
y = 3x - 9. Let's call y f(x). That means
f(x) = 3x - 9. All you have to do is apply the function f to the inequality. that is, since

-6 <= x <= 7, then

f(-6) <= f(x) <= f(7)

Apply f(x) to all three (recall that f(x) = 3x - 9), to get
3(-6) - 9 <= f(x) <= 3(7) - 9
-27 <= f(x) <= 12

But f(x) = y, so

-27 <= y <= 12

Which is the range of y.

2. To find the range of y = 2x^2 - 6x - 7, you must put this in vertex form, and the range would be dependent on the y-coordinate of the vertex. You want to put it in this form:

y = a(x - h)^2 + k

(h,k) would be the vertex, and, if "a" is positive, your range would be {x | x >= k}. If "a" is negative, your range would be {x | x <= k}.

To put y = 2x^2 - 6x - 7, you must complete the square. I'll leave all the details for you to figure out, in completing the square.

y = 2x^2 - 6x - 7
y = 2(x^2 - 3x) - 7
y = 2(x^2 - 3x + 9/4) - 7 - 9/2
y = 2(x - [3/2])^2 - 14/2 - 9/2
y = 2(x - [3/2])^2 - 23/2

Therefore, your vertex is at (3/2, -23/2). Since a = 2 (positive), your range would be {x | x >= -23/2}

3. Product of 3 consecutive whole numbers if the largest is (n+1). Since these are *consecutive*, that means they differ by 1. If (n+1) is the largest consecutive whole number, that would make the others (n+1)-1 and (n+1)-2. That is, we have

n + 1 {biggest whole number}, followed by
n
n - 1

The product of them is just that, them multiplied together.
(n + 1)(n)(n - 1).

We can rearrange this for easier multiplication. After all, multiplication is commutative.

n(n - 1)(n + 1)

Remember that difference of squares a^2 - b^2 factors into
(a - b) (a + b). Here, we have a difference of squares as well, which we can multiply out with ease.

n(n^2 - 1). Therefore, we get
n^3 - n as our final result.

4.
P(x)= x^4 - x^3 - 5x^2 + 9x - 1
D(x)= x^2 - 2x + 3
P(x) is written in the form P(x) = D(x)Q(x) + R(x)

To calculate R(x), we must use synthetic long division. What you're going to have is a quotient (the stuff on top of the division symbol), and a remainder (the stuff that remains at the bottom after performing your final subtraction during the long division).

As I've stated in almost all long-division questions I answer on here, showing you the long division on here is difficult. All I can give you is the final result, and the details are up to you to figure out.

R(x) = -4x + 5

5. Possible rational roots of 3x^4 - 4x^3 + 7x - 6.

To obtain them, you have to look at ALL the factors of the coefficient of the x term with the highest power (x^4, and the coefficient is 3), and you look at ALL the factors of the constant term (the term without an x, which is -6). Remember that you're dealing with both positive and negative factors.

Factors of 3: +/-1, +/- 3
Factors of 6: +/- 1, +/- 2, +/- 3, +/- 6

Now, what you want to do is put ALL the factors of 6 over ALL the factors of 3. [In general, all the factors of the constant term over all of the factors of the coefficient of the highest power of x]. We deal with it in an "all combinations" type manner.

Possible rational roots: +/- (1/1), +/- (1/3), +/- (2/1), +/- (2/3),
+/- (3/1), +/- (3/3), +/- (6/1), +/- (6/3).

Cleaning this up a bit, our factors are

+/- 1, +/- (1/3), +/- 2, +/- (2/3), +/- 3, +/- 1, +/- 6, +/- 2

Notice that we have some repeats as a result. We can eliminate these repeats, and our total possible rational roots, this type separating the +/- ones, would be:

1, -1, 1/3, -1/3, 2, -2, 2/3, -2/3, 3, -3, 6, -6

6. Find all the roots of 2x^3 - 5x^2 - x + 6 = 0

Possible rational roots: 6, -6, 3, -3, 2, -2, 1, -1, 3/2, -3/2, 1/2, -1/2

Let p(x) = 2x^3 - 5x^2 - x + 6.
Test p(-1): p(-1) = -2 - 5 + 1 + 6 = -7 + 7 = 0.
Therefore, since -1 is a root, (x - (-1)), or (x + 1) is a factor.

We perform long division: (x + 1) into (2x^3 - 5x^2 - x + 6). Without showing you the details (which, I repeat, is hard to show on here), you should get the result x^2 - 7x + 6.

Therefore, for 2x^3 - 5x^2 - x + 6 = 0, we can factor
(x + 1) (x^2 - 7x + 6) = 0. Factoring one more time,
(x + 1) (x - 6) (x - 1) = 0. Therefore, equating each one of those factors to 0 will give us
x = {-1, 6, 1}

7. Area of triangle formed by x-axis, y-axis, and straight line
3x + 2y - 12 = 0.

To solve this, we need to first find out what the x and y intercepts are. These would tell us the base and height respectively (since they represent the vertices of the triangle).

To solve for the x-intercept, make y = 0: Then 3x + 2(0) - 12 = 0,
3x - 12 = 0, 3x = 12, x = 4. So x = 4 is our x-intercept (and our base is 4).

For the y-intercept, make x = 0: Then 3(0) + 2y - 12 = 0,
2y - 12 = 0, 2y = 12, y = 6. So y = 6 is our y-intercept (and our height is 4).

The area of a triangle is (1/2) (base x height). Base = 4, height = 4, A = (1/2)(4)(4) = 8.

The area of the triangle is 8.

8. The object seems to be moving in a linear fashion. If we let
x = seconds and y = distance, then all we have to do is find the equation of the line that goes through (1, 64) and (2, 48).

First, we calculate the slope.

m = (y2 - y1) / (x2 - x1)
m = (48 - 64) / (2 - 1)
m = (-16)/1 = -16

To calculate the slope of our line, we use the slope formula again, but this time it would be with the points (1, 64) and (x, y). This time, we have our calculated slope m = -16, so

(y2 - y1) / (x2 - x1) = m
(y - 64) / (x - 1) = -16
y - 64 = -16(x - 1)
y - 64 = -16x + 16
y = -16x + 80

I took the wrong approach to this question and instead calculated the distance based on the second. Nevertheless, we can still calculate the distance by determine when it stops. If y = 0, that would indicate at what second it stopped.

0 = -16x + 80
-80 = -16x, so x = 5.

That means after 5 seconds, it has stopped.

What we CAN do is use sigma notation, in that we want to calculate

SUMMATION (x = 1 to 5, -16x + 80) =
(-16)SUMMATION (x = 1 to 5, x) + SUMMATION (x = 1 to 5, 80)
(-16)[1 + 2 + 3 + 4 + 5] + 80(5)
(-16)[15] + 400
-240 + 400 = 160

So the total distance is 160.

Answer 2

-7= 3x-8 when putting -8 to the other side of equation, the sign will change, so it's -7+8= 3x which means 1=3x therefore x= 1/3 and don't worry about the order, it doesn't matter! there's a rule in math about the order! you can easily check it by substituting 1/3 into the equation i.e -7= 3*1/3-8 -7=-7!!!

Answer 3

Your question is difficult. Congratulations to Puggy!