DS2011

The domain must be all real numbers except for when \(\sqrt{6-x-x^2-2x^2}=0\) Simplifying this, we get:

\(-3x^2-x+6=0\)

a = -3, b = -1, and c = 6.

\(x = {1 \pm \sqrt{1+72} \over -6}\)

\(x=\frac{1\pm\sqrt{73}}{-6}\)

The domain of the function \(f(x)=\sqrt{6-x-x^2-2x^2}\) is all real numbers except \(x=\frac{1+\sqrt{73}}{-6}\) and \(x=\frac{1-\sqrt{73}}{-6}\).

Given:

AC is the diameter of a circle \(\omega\)

circle \(\omega\) has radius 1

D is a point on AC such that CD = 1/5

B is a point of circle \(\omega\) such that DB is perpendicular to AC

E is the midpoint of DB

Question:

AE = ?

I will assume that B is one the circumference of the circle, or else, it is not possible to find AE.

Using https://jspaint.app/#local:cf1101d801c0e, we can draw this figure:

We can calculate length AE by doing AD^2 + ED^2. 

We know that AD is 2 - 1/5 = 1 4/5, and squaring that gives us 3.24, so we get:

3.24 + ED^2 = AE^2

To calculate length ED, we first need to find length BD, which can be found by drawing a line OB, and then using the Pythagorean Theorem:

OD^2 + BD^2 = OB^2

We know that OD is 4/5 and OB is 1, so we can solve for BD:

0.8^2 + BD^2 = 1

0.64 + BD^2 = 1

BD^2 = 0.36

BD = square root of 0.36 or sqrt 36/100 = 6/10 = 0.6

BD = 0.6, so the midpoint divides BD into 2 equal segments, which as equal to 0.6/2 = 0.3, so ED = 0.3.

Using the Pythagorean Theorem to find AE, we get:

AD^2 + ED^2 = AE^2

1.8^2 + 0.3^2 = AE^2

3.24 + 0.09 + AE^2

AE^2 = 3.33

AE = square root of 3.33 or approximately 1.825.

The answer is \(\sqrt{3.33}\) or 1.825

Answer: \(\sqrt{3.33}\) or \(1.825\)

Okay we got the same answer.

Thank you!

Simplify:

\(x^2+3x+5=0\)

Plugging in the values a = 1, b = 3, and c = 5 into the quadratic equation \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\), we get:

\(x = {-3 \pm \sqrt{9-20} \over 2}\)

The discriminant, or \(\sqrt{b^2-4ac}\), is less than 0, meaning that this quadratic equation has complex roots, which include imaginary numbers, as we are dealing with the square root of a negative number.

Further simplifying, we get:

\(x = {-3 \pm \sqrt{11} i \over 2}\), where i is not in the square root, but rather it is the square root of 11 multiplied by i.

We get that the complex roots of the quadratic equation are:

\(x = \frac{-\sqrt{11}i-3}{2}\), \(x = \frac{\sqrt{11}i-3}{2}\)

So, we get that a = \( \frac{-\sqrt{11}i-3}{2}\), and b = \(\frac{\sqrt{11}i-3}{2}\).

Plugging in these values into the expression \(\frac{a^2}{b} + \frac{b^2}{a}\), we get:

\(\frac{ (\frac{-\sqrt{11}i-3}{2})^2}{ \frac{\sqrt{11}i-3}{2}} + \frac{( \frac{\sqrt{11}i-3}{2})^2}{ \frac{-\sqrt{11}i-3}{2}}\)

Can you do the rest?

Simplify:

3x^2 - 11x - 7 = 0

Finding the roots by plugging in the values a = 3, b = -11, and c = -7 into the quadratic equation \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\), we get:

x = \(\frac { \sqrt{205} + 11}{6}\), x = \(\frac { -\sqrt{205} + 11}{6}\)

We now have that a = \(\frac { \sqrt{205} + 11}{6}\) and b = \(\frac { -\sqrt{205} + 11}{6}\).

Plugging in these values for the expression \(\frac{1}{a+7}+\frac{1}{b+7}\), we get:

\(\frac{1}{\frac { \sqrt{205} + 11}{6}+7}+\frac{1}{\frac { -\sqrt{205} + 11}{6}+7}\)

Simplifying this expression, we get:

\(\frac{1}{\frac { \sqrt{205} + 53}{6}}+\frac{1}{\frac { -\sqrt{205} + 53}{6}}\)

Removing the 6 and multiplying 1 by the reciprocal of \({\frac { \sqrt{205} + 53}{6}}\), we get:

\(\frac{6}{ \sqrt{205} + 53}+\frac{6}{ -\sqrt{205} + 53}\)

We can plug each of these terms in the calculator to get

\(0.089 + 0.155\)

Adding these two values, we get:

\(0.244\)

Answer: 0.244

\(f(x) = \left\lfloor\frac{2 - 3x}{3x + 1}\right\rfloor\)

When x = 1, f(1) = -0.25 rounded down = -1

When x = 2, f(2) = -4/7 rounded down = -1

When x = 3, f(3) = -0.7 rounded down = -1

Let's try bigger values of x to see if the answer is less than -1.

When x = 33, f(33) = -0.97 rounded down = -1.

For any x value, the answer rounded down will be -1, so we get:

-1 -1 -1 -1 -1 - ... -1 -1 = -1000

Answer: -1000

Oh yes I did not see that A had to be positive.

We do not know the elevation of Boston going from a cafe to a boba shop, so we can not find the answer. Although, the 2 possible answers are:

(100 + 150) / 2 = 125 meters per hour, because if you go uphill or downhill, the same route in reverse is the other, ex if going downhill, the reverse is uphill.

120 meters per hour

It's 120 meters per hour or 125 meters per hour.

The equation of this line in the form y = mx + b is y = (1/2)x + 2, so we can do:

2y = x + 4

-x + 2y = 4

Answer: -x + 2y = 4

60 - 24 = 36, so we can divide 36 by 5 because we have to add a number 5 times, and we get 7.2, so the arithmetic sequance is growing by 7.2 each progressive term.

24, 24 + 7.2, 24 + 14.4, 24 + 21.6, 24 + 28.8, 60

a = 31.2

b = 38.4

c = 45.6

d = 52.8

a + b + c + d = 168

Answer: 168

We are adding by 7 for each progressive term, so we can subtract 97 by 13 and divide the amount by 7 and after add 1 because we have to count the first term as well to get the amount of integers.

[(97 - 13) / 7] + 1

(84 / 7) + 1

12 + 1

13

Answer: 13

For there to be 1 solution, the discriminant must be equal to 0, so b^2-4ac = 0

Simplify:

\(2x^2-x-jx+3=0\)

a = 2, b = -1, and c = 3

Pluggng a and c in, we get:

\(\sqrt{b^2-24}=0\)

This means that \(b^2 = 24\).

\(b=\pm \sqrt 24\)

\(b=\pm4.89\)

We already have -1, and we are subtracting j, so we have:

-1 - j = 4.89

-j = 5.89

j = -5.89

and,

-1 - j = -4.89

-j = -3.89

j = 3.89

the values of j are -5.89 and 3.89

Answer: -5.89, 3.89

It can not be a, because that is an exponential function, and the range gets exponentially larger as x gets larger. 

It can not be b, because a in a  parabola, there is a vertex, which means that there is a minimum or maximum value to y.

I haven't yet learned about logarithms, but I can assume that the range is not equal to the domain there.

It is d, because the range is eqqual to the domain, since y = x and y is the range and x is the domain.

Answer: D

My idea: Reflect triangle XYZ across line segment YZ, to form a square, which is possible because this is a 45 45 90 triangle where YZ is the hypotenuse. We can find the area of the semicircle by finding the area of the circle and then dividing it by 2.

The diameter of the circle will be 10 because it is inscriibed in the square of length 10, so the radius is 5.

using A = pir^2, we get 25 pi, or 12.5 pi for the semicircle.

We can approximate 12.5pi as 36.27

Answer: 12.5pi or 36.27

same amount of members, but the amount of wall is 4.8 times more, so 50 x 4.8 = 240 minutes

Answer: 240 minutes

                                                                                                           BLUE

                                                                                                             45

                                                                            White & Blue                               Blue & Green

                                                                                    18                                                28

                                                                                                White, Green, & Blue

                                                                                                               5

                                                                WHITE                         White & Green                       GREEN

                                                                   57                                       41                                      62

Sorry I could not find a good Venn Diagram maker and do it there to upload an image, but this is my best without taking a lot of time.

Just imagine circles around the WHITE, GREEN, and BLUE and the places where they intersect.