AzizHusain

lxl + lyl = 1

We need to get rid of the two absolute value signs from x and y:

Since we have two variables and each absolute value has 2 possiblities, we would end up with a total of 4 equations [abs(x) = x / -x & abs(y) = y / -y]:

x + y = 1, x - y = 1, -x + y = 1, & -x - y = 1

1) x + y = 1 --> y = -x + 1 --> Notice, we took positive x and y, so we are interested in values: x and y are >= to 0. If you graph this, you get a line that is only in the 1st quadrant.

2) x - y = 1 --> y = x - 1 --> We took positive x and negative y, so we are interested in values: x is >= 0 and y < 0. If you graph this, you get a line that is only in the 4th quadrant.

3) -x + y = 1 --> y = x + 1 --> Follow the pattern...negative x and positive y means the 2nd quadrant.

4) -x - y = 1 --> y = -x - 1 --> Follow the pattern...negative x and negative y means the 3rd quadrant.

Notice that each of the 4 lines has a slope of 1 or -1.

Now, if you graph each of these, you would find that it looks like the graph Phil posted of a square that has sides of sqrt(2).

Have fun and enjoy your time there, Rosala!

4x^2 - 7x - 2 = 0

Use the quadratic formula: x = (7 +/- sqrt[49 - 4(4)(-2)]) / 2(4) -->

(7 +/- sqrt(49 + 32)) / 8 -->

(7 +/- sqrt(81)) / 8 -->

(7 +/- 9) / 8 -->

x = (7 + 9) / 8 = 16 / 8 = 2

&

x = (7 - 9) / 8 = -2 / 8 = -1/4 = -0.25

Note: 4(2^2) -2 ?= 7(2) --> 16 - 2 = 14 --> Yes, therefore, x = 2 is a solution.

4((-1/4)^2) - 2 ?= 7(-1/4) --> -1.75 ?= -1.75 --> Yes, therefore, x = -1/4 is a solution, too. 

Each term has 3v^2 available:

3v^2(3v^2 + 8v + 4)

Next, we factor out the (3v^2 + 8v + 4):

(3v^2 + 8v + 4) -->

(3v + 2)(v + 2)

Putting it all together, we have:

3v^2(3v + 2)(v + 2)

Cross multiply:

-30x = -20(x-2) 

-30x = -20x + 40

-10x = 40

x = -4

Compare the cross or diagonal terms first.

9x^5*y^4 & 3x --> we can cross out the 3x entirely and change the 9x^5*y^4 into 3x^4*y^4 by dividing 9 by 3 and crossing out one x in x^5.

4y & 2y --> we can change 4y into 2(2y) and cross out the 2y.

Notice by crossing out the 3x and 2y, we are left with 1 / 1.

Thus, we have: 3x^4*y^4 / 2  * 1 / 1 -->

3x^4*y^4 / 2.

Let's try simplifying each numerator/denominator:

x - 1 is already simplified.

4x - 8 --> 4(x - 2).

x^2 + x - 6 --> What 2 numbers add up to 1 and multiply to give -6? --> (x + 3)(x - 2).

x + 1 is already simplified.

Now, notice we can cross out the (x-2)'s and we end up with:

(x - 1) / 4 * (x + 3) / (x+1) -->

Now, just multiply like normal: (x - 1)(x + 3) / 4(x + 1) -->

Perform the FOIL method for the numerator and distribute the denominator -->

(x^2 + 2x - 3) / (4x + 4)

3v^3*y^4 - 48v^3

1. We can factor out 3v^3 from both terms -->

3v^3(y^4 - 16)

2. We can further factor out (y^4 - 16) -->

3v^3(y^2 + 4)(y^2 - 4)

3. Finally, we can factor out (y^2 - 4) -->

3v^3(y^2+4)(y + 2)(y - 2)

sqrt(96) = sqrt(16*6) = sqrt(16)*sqrt(6) = 4*sqrt(6).