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# Algebra/Algebra 2 help

Question
hi i need a little help on a few questions i hope there not too many

Assume y varies inversely as x. If y = -32 when x = 2, what is x when y = 4?
-12  -14
-18  -16

7. Find all the roots of the equation.

x^3 - 2x^2 - x + 2 = 0

No roots  (0, -1)
(-1, 1,2 ) (-1)

8. Find all zeros of the polynomial

x^3 - 5x^2 + x - 5

5, i  -5, -i
0, 1  5, ± i

9. Find the coordinates of Q given that M is the midpoint of line segment PQ:

P(6, -2) and M(0, 5)

(4, 8)  (6, -12)
(-6, 12)  (12, -6)

10. Find the midpoint of the segment joining the two points:

(0.5, -1) and (-1, 1)

(-0.25, 0)  (0, -0.25)
(0.25, 0)  (0, 0.25)

11. Find an equation of the circle with the center and radius:

(-3, 1); r = 5

(x - 3)2 + (y + 1)2 = 25  (x + 3)2 + (y - 1)2 = 25
(x - 3)2 + (y + 1)2 = 5  (x + 3)2 + (y + 1)2 = 5

12. If the directrix of a parabola is D: x = -7 and the vertex is V(1, 3), what is the focus?
(-10, 5)  (9, 3)

(3, 5)  (-3, 10)

Solve by completing the square:
x2 - 4x = -3

4, 2  1, 3
2, 3  0, 1

14. Identify the conic section whose equation is given:
5y2 + 50y + 275 = 100x - 2x2

Circle  Parabola
Hyperbola  Ellipse

Choose the correct foci:

y2 = 5x2 + 25

(0, ±5.48)  (0, 7.75)
(±30, 0)  (0, ±30)

Solve:

c3 + c2 - 7c - 3 = 0, given root -3

c = 3 or -1  c = -3, -0.41, 2.41
c = 1  c = -1

17. Find the length of the legs of a right triangle having perimeter 56m if the hypotenuse is 25m. 12m and 14m  10.5m and 16m
17m and 20m  7m and 24m

18. Find the dimensions of a rectangle that has area of 10 and a diagonal of length 5. 2.24 x 6.32  2.24 x 4.47
2.24 x 4.91  2.24 x 7.07

19. A 20 m ladder and a 15 m ladder were leaned against a building.  The bottom of the longer ladder was 7 m farther from the building than the bottom of the shorter ladder, but both ladders are the same distance up the building.  Find the distance. 12m  16m
14m  10m

20. The area of a rectangle is 48 m2. The length of a diagonal is 10 m. Find the perimeter of the rectangle. 28m  14m
42m  80m

21. A number y varies jointly as x and z. If y = -24 when x = 4 and z = 3, what is y when x = -6 and z = -2? -20  -32
-16  -24

i hope you can help me out i have all the rest, these are just giving me alot of trouble

Assume y varies inversely as x. If y = -32 when x = 2, what is x when y = 4?
This says xy = -64; solve if y = 4.

7. Find all the roots of the equation x^3 - 2x^2 - x + 2 = 0.
That is (x-2)(x^2-1), and (x^2-1) is (x+1)(x-1); from here, 3 roots can be found.

8. Find all zeros of the polynomial x^3 - 5x^2 + x - 5.
That factors into (x-5)(x^2+1).  Clearly the right choice has a 5 and a couple of imaginary numbers.

9. Find the coordinates of Q given that M is the midpoint of line segment PQ P(6, -2) and
M(0, 5).  If x is at 6, 0 is the midpoint, then -6 is at the other end.  If -2 is at the end, 5 is the midpoint, the difference is 7.  The other points is then at 5+7.

10. Find the midpoint of the segment joining the two points (0.5, -1) and (-1, 1).
The two x terms are 0.5 and -1, so find the average for the x midpoint.
The two y terms are -1 and 1, so the average of those two is clearly 0.

11. The equation for a circle is (x-x0)^2 + (y-y0)^2 = r^2 where
(x0,y0) is the center and r is the radius.

12. If the vertex is at (a,b) and the directrix is at x=c, the focus is the same distance the other direction.  That is, at (a+(a-c),b).

13. Completing the square of x^2 - 4x = -3 means adding 4 to both sides.
This gives x^2 - 4x + 4 = 1.  That is the same as (x-2)^2 = 1, so (x-2)=1 or (x-2)=-1.

14. Take 5y^2 + 50y + 275 = 100x - 2x^2 and change it to 5y^2 + 50y + 2x^2 - 100x = -275.
That factors to 5(y^2 + 10y) + 2(x^2-50x) = -275.

Completing the y square means adding (10/2)^2 = 25 to the inside, and since that is times 5, that is adding on 125 on the lest, so add 125 to the other sides as well.

Completing the square on x means adding (50/2)^2 = 625 on the inside, and this is times 2,
so add 2*625 = 1250 to the other side as well.

This gives 5(y+5)^2 + 2(x-25)^2 = 1100.

That is an elliptic equation.

15. y2 = 5x2 + 25

(0, ±5.48)  (0, 7.75)
(±30, 0)  (0, ±30)

16. Solve:  c3 + c2 - 7c - 3 = 0, given root -3

c = 3 or -1  c = -3, -0.41, 2.41
c = 1  c = -1

17. This means the legs have to add to 31.

18. Find the dimensions of a rectangle that has area of 10 and a diagonal of length 5.
Doing multiplication of the 1st choice is clearly at least 12, so that's wrong.
For the last choice, it is at least 14, so that's wrong.  That leaves 2.24 x 4.47
or 2.24 x 4.91.  The first is roughly 2 1/4 x 4 1/2 = 9/4 x 9/2 = 81/8, which is pretty close to 10.  The second is roughly 9/4 by 0.9 less than 5, which is 45/4 - 4.5, so clearly that is wrong as well.

19. A 20 m ladder and a 15 m ladder were leaned against a building.  The bottom of the longer ladder was 7 m farther from the building than the bottom of the shorter ladder, but both ladders are the same distance up the building.  Find the distance.

From this, we know the height of both is y.  For the 1st triangle, the base is x and the hypotenuse is 15.  For the 2nd, the base is x+7 and the hypotenuse is 20.
This gives us the equations 225 = x^2 + y^2 and 400 = (x+7)^2 + y^2.

Multiply out (x+7)^2, subtract the 1st equation from the 2nd equation, and then x can be solved for.  Once this has been done, y can be found.

20. If xy = 48 and x^2 + y^2 = 100, the first thing to try is x=6 and y=8,
since 6^2 + 8^2 = 36 + 64 = 100 = 10^2.  This gives the right area, so the perimeter is
6+6+8+8=...

21. If x varies by -3/2 and z varies by -2/3, then y varies by the product of -3/2 * -2/3.
Isn't the answer to that y doesn't vary?

Algebra

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#### Scott A Wilson

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