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By geometric series, b = 50/7.

If we simplify and use special products. and then combine like terms, and then you'll get 2x^2 + 22x + 65. The equation for solving the axis of symmetry is -b/2a In ax^2 + bx + c. I think you can do the rest. 

^0.5 means ^(1/2), and that basically means square root of. You can do the rest. Just remember to solve the parenthesis first

The set is (-2, 4) 

2*2 = 4. 4+2 = 6

124556 ( 360 ) , 134455 ( 180 ) , 222556 ( 60 ) , 223455 ( 180 ) , >>Total combinations = 4

>>Totals in parentheses = 780 permutations

Let a be the smallest integer satisfying the inequality x^2 - 15 < 2x, and let b be the largest integer satisfying the same inequality. What is b-a?

\(x^2 - 15 < 2x\\ x^2 - 15 -2x <0\\ x^2 -2x - 15 <0\\\)

This is a concave up parabola

It will be less than 0 between the roots

the roots are

\(x = {2 \pm \sqrt{4+60} \over 2}\\ x = {2 \pm 8 \over 2}\\ x=1\pm4 \)

the equation is satified for all x values between -3 and 5  BUT  NOT inclusive of -3 and 5)

So the smallest integer is a=-2   and the largest is b=4

4 - - 2 = 6

This is just asking fo whole numbers between -2 and 4     (NOT including -2 or 4)

\(Let\;\;f(x)=\lfloor x\lfloor x \rfloor \rfloor \quad for \;\;x\ge 0\\ (a)\;\; find\;all\;x\ge0\;\;such\;\;that\;\; f(x)=1\)

If  0<=x<1 then  f(x)=0

If  1<=x<2 then   f(x)= floor of (x*1) = 1

If x>=2      then    f(x)>1

so      \(if\;\; 1\le x<2 \;\;then \;\;f(x)=1\)

(b)   

\(Let\;\;f(x)=\lfloor x\lfloor x \rfloor \rfloor \quad for \;\;x\ge 0\\ (a)\;\; find\;all\;x\ge0\;\;such\;\;that\;\; f(x)=3\)

\( let\;\; 0\le\delta <1\\ If\;\;x=2+\delta \;\;then\\ \lfloor x\lfloor x \rfloor \rfloor=\lfloor (2+\delta) 2) \rfloor=\lfloor (4+2\delta) \rfloor=4\;\;or\;\;5 \)

So if  x is 1 and a bit then f(x)=1

If x=2 and a bit then f(x)=4 or 5

f(x) cannot equal 3

You can think about the rest yourself.

The floor function means round down to the nearest integer

The ceiling function means round up to the nearest integer.

For example:

\( \lfloor -4 -.5 \rfloor =\lfloor -4.5 \rfloor =-5\\ \lceil 4 +.5 \rceil =\lceil 4.5 \rceil =5\\ \)

Can you take it from there?

First, you need to determine when Ben will reach his minimum height

\(\frac{5}{(2)(1)}\) = 2.5 seconds

Then, plug 2.5 into the equation as t

(2.5)² - 5(2.5) + 29

Simplify

6.25 - 12.5 + 29

Simplify

22.75

Rounded to the nearest foot, his minimum height will be 23 feet

I actually found a similar question back in 2020 that Melody answered, so I'm just going to reuse her method (thank you Melody!)

You can use the Tangent-Secant Power Theorem for this problem.

"If a tangent and a secant intersect in the exterior of a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant and its external secant segment."

XY² = XB(AX)

Plug in your numbers

(8)² = XB(16)

Simplify

64 = 16XB

Divide by 16 on both sides

4 = XB

Now, to find AB, you subtract XB from AX

AB = 16 - 4

AB = 12

Hope that helps!

Let x = the number of hours Naomi worked

Let y = the number of hours Hudson worked

Using the numbers they gave us regarding how many shirts each of them can iron per hour, you multiply the number of shirts Naomi can iron per hour by x (the number of hours Naomi worked), and vice versa for Hudson. Combined, they ironed 395 shirts. That is what it will be equal to.

Combined, they worked 13 hours. You dont know how many hours they each worked, so it becomes x + y = 13.

Here is your system of equations:

35x + 20y = 395

x + y = 13

This question appears to be very simple, have you tried drawing the diagram?

Let ABCD be a parallelogram. If AC = 9 and BD = 17, then find AB^2 + BC^2.

\(BC^2=4.5^2+8.5^2-(2*4.5*8.5*cos\alpha) \qquad (1)\\~\\ AB^2=4.5^2+8.5^2-(2*4.5*8.5*cos(180-\alpha))\\ AB^2=4.5^2+8.5^2+(2*4.5*8.5*cos(\alpha))\qquad (2)\\ (1)+(2)=\)

You can check and finish it 

Let's say they delivered x newspapers in total,

     so Ben delivered 5/8x newspapers

          Katie delivered 3/8x newspapers

      we know Katie delivered 8 fewer than Ben

               so 5/8x - 3/8x = 8

                              2/8x = 8

                              1/4x = 8

                                  x = 32
                            so 5/8 * 32 = 20

                            so Ben delivered 20 newspapers

let 6x be cookies,

  doughnuts = x

  muffins = 3x

 6x = 42

  x + 42/6 = 7 

  x = 7

  3x = 3 * 7 = 21

  6x = 42

Total dessert

= 7 + 21 + 42

= 70

{Drew played 11/12 of the games, Will played 4 games.}

1 - 11/12 = 1/12

4 / 1/12 = 4 * 12 = 48

48 * 11/12 = 44

The ratio of total desserts to cookies is \(10: 6 \). 

We know that there are \(6 \times 7 = 42 \) cookies. 

Now, we do the same thing to the desserts side. 

Can you take it from here?

Let Chase had jellybeans = x

Peter had jellybeans = 5/6 of x

                                  = 5x/6

Chase gave jellybeans to Peter = 1/2 of x

                                               = x/2

Now Chase had jellybeans = x - x/2

                                           = x/2

Now Peter had jellybeans = 5x/6 + x/2

According to problem

                5x/6 + x/2 = x/2 + 55

                5x/5 + x/2 - x/2 = 55

                     5x/6 = 55

                          x = (55 * 6)/5

                          x = 66

Chase had jellybeans at first = 66

Because there are absolute value bars inside, which makes it positive. Likewise you don't see a negative sign behind the square root I put. Teachers will remember to test you so have to remember to only use the plus or minus in case of quadratics or something like that. But to put simply square root itself must be positive in the real domain. It doesn't feel right to just say that the sqrt(9) = -3.

no I think it's 1