Logarhythm

:D 

First, simplify the equation:
\(3x + \frac{2}{x} - 4 = 2(x + 9) - (7 - \frac{1}{x})\)

\(3x + \frac{2}{x} - 4 = 2x + 18 - 7 + \frac{1}{x}\)

\(x - 15 = -\frac{1}{x}\)

\(x^2 - 15x = -1\)

\(x^2 + 1 = 15x\)

Now plug it into \(\frac{x^2 + 1}{x}\):

\(= \frac{(15x)}{x}\)

\(= 15\)

\(\frac{x+4}{x+5} = \frac{x-5}{2x}\)

2x(x + 4) = (x - 5)(x + 5)

2x² + 8x = x² - 25

x² + 8x + 25 = 0

\(x = {-8 \pm \sqrt{8^2-4(1)(25)} \over 2(1)}\)

\(x = {-8 \pm \sqrt{-36} \over 2}\)

\(x = {-8 \pm 6i \over 2}\)

\(x = -4 \pm 3i\)

Which means the solutions are: 

x = -4 + 3i

x = -4 - 3i

First let's simplify the expression:
-2x² - 20x + 14x - 53 + 17 

= -2x² - 6x - 36 

Now use completing squares to change it into the form a(x + d)² + e:
\(-2x^2 - 6x - 36\)

\(= -2(x^2 + 3x + 18)\)

\(= -2(x^2 + 3x + 2.25 - 2.25 + 18) \)

\(= -2((x + 1.5)^2 + 15.75)\)

\(= -2(x + 1.5)^2 - 31.5\)

a + d + e = -2 + 1.5 + (-31.5)

               = -0.5 - 31.5

               = -32 

since the angle is 60, we know it's a 30 60 90 triangle. 

this means her height is 39 ÷ √3 = 13√3 ≈ 22.52 inches. 

16 ÷ 1.5 = \(10 \frac{2}{3}\)

15 ÷ 1.5 = 10

12 ÷ 1.5 = 8

Let's say the bottom side is 16 x 15. (it won't matter)

since 10 cubes can fit in 16 and 15, there will be 10 x 10 = 100 cubes. 

Then 12 is the height and 8 cubes can fit. 

This means 100 x 8 = 800 cubes. 

(so it's just 10 x 10 x 8)

It's okay I figured it out :)
Thank you so much for answering nonetheless

perimeter = 2 x length + 2 x width   or  2(length + width)

and since the perimeter is 24 inches:
24 = 2 x length + 2 x width

12 = length + width

diagonal length = \(\sqrt{l^2 + w^2}\), when l = length and w = width, according to pythagoras' theorem.

since they are both squared, we want a big number (example: 2² + 3² < 1² + 4²)

this means to maximize the diagonal, the length is 11 and width is 1. (11 + 1 = 12)

\(\sqrt{11^2 + 1^2} = \sqrt{122} ≈ 11.06\)

hope this helps; tell me if I did anything wrong :)

c(2,12,3) = \(\frac{2}{12} + \frac{12}{3} + \frac{3}{2}\)

               = \(\frac{1}{6} + \frac{24}{6} + \frac{9}{6}\)

               = \(\frac{34}{6}\)

               = \(\frac{17}{3} \) or \(5\frac{2}{3}\)

Hi, thanks for answering but I'm not sure that's how my teacher wants us to do it...

She says we have to put this on a number line and write out the cases, then solve it.

Not sure if you'll see this post again but it's fine; I'll just ask in class.

Thank you so much though :)