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First, we need to isolate the square root on the left-hand side of the equation. We can do this by squaring both sides of the equation. This gives us:

9x^2-4x+4=(2(3x-2)+sqrt(x))^2

Expanding the right-hand side of the equation gives us:

9x^2-4x+4=12x^2-12x+4

Subtracting 12x^2-12x+4 from both sides of the equation gives us:

-3x^2+8x-0=0

Factoring the left-hand side of the equation gives us:

-(x-2)(3x-0)=0

This gives us two possible solutions for x:

x=2 x=0

However, the value of x=0 is not a valid solution, because it would make the square root on the left-hand side of the original equation undefined. Therefore, the only valid solution is x=2.

p = 11.

Let the total number of people at the Math convention==P

7 / 12 P - percentage of people bored by cable news

7/12P ==280

P==280 / (7/12) ==280 x 12/7 ==480 people at the Math convention.

480  -  280 ==200 people who are not bored by cable news.

Real numbers a and b satisfy  

a + ab = 250  

a - ab = -240  

Enter all possible values of a, separated by commas.  

If you add the two equations together                a + ab  =   250  

                                                                           a – ab  = –240  

                                                                          ————————  

you get                                                              2a         =  10  

from which                                                                  a  =  5  

The problem doesn't ask for b,  

but ... find b by substituting  

a back into a + ab = 250                                    5 + 5b  =  250  

                                                                                5b  =  245  

                                                                                  b  =  49  

check b by substituting  

a back into a – ab = –240                                  5 – 5b  =  –240  

                                                                              –5b  =  –245  

                                                                                  b  =  49  

_.

I don't think UV=r because U is outside the circle. 

Thanks, that's right!

For 3+x/5+x and 1+x/2+x to be equal, we need to equate the expressions.

3+x/5+x = 1+x/2+x

Combining like terms, we get:

13x/10 = 3+x/2

Multiplying both sides by 10, we get:

13x = 30+5x

Subtracting 5x from both sides, we get:

8x = 30

Dividing both sides by 8, we get:

x = 30/8

Simplifying, we get:

x = 15/4

Therefore, the value of x for which 3+x/5+x and 1+x/2+x will be equal is 15/4.

[14 C 5]  +  [14 C 4] ==2,002  +  1,001 ==3,003 solutions.

One ordered pair (a,b) satisfies the two equations ab^4 = 48 and ab = 72. What is the value of b in this ordered pair?  

To find b, consider                                                  ab⁴  =  48  

We will divide both sides by ab.  

Since ab=72, we will divide the left side  

by "ab" and the right side by its equal 72.  

                                                                              ab⁴         48  

                                                                             ——   =   ——  

                                                                              ab           72  

Note that ab⁴ = (ab) * (b³)  

Cancel ab out of the left side.  

Reduce 48/72 on the right side.  

                                                                               b³           2  

                                                                             ——   =   ——  

                                                                                1            3  

                                                                                 b   =   cube root of (2 / 3)  

_.

1994  -  1990 ==4 years' period involved.

[1.5 / 1.4]^(1/4) ==1.017397 - 1  x  100==1.7397% - average annual compound rate of change

The probability that the first card Grok draws has an even number is 3/4. There are 3 even numbered cards (2, 4, and 6) and 7 total cards, so the probability of drawing an even number is 3/7.

The probability that the second card Grok draws has a multiple of 3 is 2/4. There are 2 multiples of 3 (3 and 6) and 4 total cards remaining, so the probability of drawing a multiple of 3 is 2/4.

The probability that both events happen is the product of the probabilities of each event happening. In this case, the probability is 3/4 * 2/4 = 3/8.

Therefore, the probability that Professor Grok draws two cards from Ms. Q's deck at random without replacement, where the first card has an even number and the second card has a multiple of 3 is 3/8.

Thanks for the help! Unfortunately, 1/7 isn't correct.

The probability of event A happening, then event B, is the probability of event A happening times the probability of event B happening given that event A already happened. In this case, event A is drawing an even number and event B is drawing a multiple of 3.

There are 3 even numbers in the deck (2, 4, and 6) and 3 multiples of 3 (3, 6, and 9). So, the probability of drawing an even number is 3/7.

Once Grok has drawn an even number, there are only 2 multiples of 3 left in the deck (6 and 9). So, the probability of drawing a multiple of 3 given that Grok has already drawn an even number is 2/6.

Therefore, the probability of Grok drawing an even number, then a multiple of 3 is 3/7 * 2/6 = 1/7.

Let O be the center of the circle, and let r be the radius of the circle.

Since line UV is tangent to the circle at point V, we know that UV=r.

We also know that UX=r+x and UY=r+y, where x and y are the distances from points U and V to the line XY, respectively.

Since XY is a straight line, we know that x+y=r.

Substituting r+x for UX and r+y for UY in the equation UV2=UX∗UY, we get:

(r)2=(r+x)(r+y)

r2=r2+rx+ry+xy

0=rx+ry+xy

Since x+y=r, we can substitute r for x+y in the equation 0=rx+ry+xy to get:

0=r(r−y)+xy

0=r2−yr+xy

We can factor the equation 0=r2−yr+xy to get:

0=(r−y)(r+y)

Since x+y=r, we know that y=r−x. Substituting r−x for y in the equation 0=(r−y)(r+y), we get:

0=(r−(r−x))(r+(r−x))

0=(2r−x)(2r−x)

0=(2r−x)2

Since the square of a real number is always non-negative, we know that (2r−x)2=0. Therefore, 2r−x=0.

Solving for x, we get:

x=2r

Substituting 2r for x in the equation UV2=UX∗UY, we get:

UV2=(r+2r)(r+2r)

UV2=4r2

Therefore, UV2=UX∗UY.

Could you tell me why we need ax² + (b+11)x + 8 = 0 to have equal roots at x = 2?

A bag contains yellow, green, blue, and white marbles.  The ratio of yellow, green, blue, and white marbles is 1:2:3:4.  If the bag contains 180 marbles, then how many blue marbles are there?  

Consider:  "The ratio of yellow, green, blue, and white marbles is 1:2:3:4."  

Call the number of yellow marbles X.  

So the number of green marbles is 2X. 

And the number of blue marbles is 3X.  

And the number of white marbles is 4X.  

Add up all the marbles and you have     10X  =  180  

Therefore                                                   X  =  18 yellow marbles  

                                                                 2X  =  36 green marbles  

                                                                 3X  =  54 blue marbles  

                                                                 4X  =  72 white marbles  

_.

Answered yesterday.

There's a mistake in the answer given above, the slope of the parabola should be 2ax + b, not 2a + b.

That leads to incorrect values for a and b.

Here's a solution not involving calculus.

To find out where the straight line and the parabola intersect equate the two expressions for y,

\(-11x-7=ax^{2}+bx+1,\\ \text{so} \\ ax^{2}+(b+11)x+8=0.\)

We need this to have equal roots at x = 2.

That gets you the two equations

\(-(b+11)/2a=2 \\ \text{and} \\ (b+11)^{2}-32a=0,\)

which can be solved simultaneously for a and b.

(a = 2, b = -19)

Thank you so much for your thorough answer. Could you tell me how to solve this problem without using calculus?

The line y = –11x – 7 is tangent to the parabola y = ax² + bx + 1 at point P. This means that the line and the parabola share a common point, and the line's slope is equal to the parabola's slope at that point.

The slope of the line y = –11x – 7 is –11. The slope of the parabola y = ax² + bx + 1 at point P is equal to 2a + b.

Since the line and the parabola share a common point, and the line's slope is equal to the parabola's slope at that point, we have:

-11 = 2a + b

We are given that the x-coordinate of P is –2. This means that the point P is (–2, y).

We can substitute this point into the equation of the parabola to find the value of y:

y = a(–2)² + b(–2) + 1

y = 4a – 2b + 1

We are given that the line y = –11x – 7 is tangent to the parabola at point P. This means that the point P lies on the line y = –11x – 7.

We can substitute the point P into the equation of the line to find the value of y:

y = –11(–2) – 7

y = 15

We now have two equations with two unknowns:

-11 = 2a + b

15 = 4a – 2b

We can solve these equations for a and b.

Adding the two equations, we get:

4 = 6a

a = 2/3

Substituting a = 2/3 into the first equation, we get:

-11 = 4(2/3) + b

b = -19/3

Therefore, a + b = 2/3 + (-19/3) = -17/3.

So the answer is -17/3

I recamend Symbol Lab. Type it in then find what you're looking 4

https://www.symbolab.com

Algerba Help: 

https://www.algebra.com/algebra/homework/Permutations/Permutations.faq.question.390865.html#:~:text=That%20means%20there%20are%20a%20total%20of%2090000,5%20digit%20numbers%20with%20at%20least%20one%20zero.

Gmatclub:

https://gmatclub.com/forum/how-many-5-digit-numbers-have-at-least-one-zero-digit-411268.html

Same question.

Overview:

(9*10*10*10*10)-9^5= 30951

Total 5 digit numbers-5 digit numbers without a 0=5 digit numbers with a 0

Answer is 30951

Let the number ==N

5N / [7 + N] =0

5N = 0

N = 0

Read the question, it says that five times a number IS DIVIDED BY 7 more than the number, it doesn't say that five times a number is equal to 7 more than the number.

sorry! I read 1/2 instead of 1/6.