hectictar

The literal interpretation of this:    -790 +- √790^2 - 4 (-60)(-1000) / 2(-60)

is this:     \(-790\pm\sqrt{790}^2-\frac{4(-60)(-1000)}{2}(-60)\)

However, I think what you probably mean to say is this:    [  -790 ± √[ 790^2 - 4 (-60)(-1000) ]  ] / [ 2(-60) ]

which is:     \(\frac{-790\pm\sqrt{790^2-4(-60)(-1000)}}{2(-60)}\)

The only way that a calculator would know that  -790  is part of the numerator is if you put parenthesees around the entire numerator. The same goes for the denominator.

Now because of the  ±  , to enter this expression into a normal calculator we have to enter both expressions separately.

The first expression is:  (-790+sqrt(790^2-4(-60)(-1000)))/(2(-60))

The second expression is:     (-790-sqrt(790^2-4(-60)(-1000)))/(2(-60))

I took out the spaces, changed the brackets to parenthesees, and wrote  sqrt  instead of  √  so that you can select each expression and copy and paste it directly into the calculator on this website.

For the first one you should get approximately  1.419

For the second one you should get approximately  11.748

Yep! but be careful because notice that these aren't "regular" coordinates, they're polar coordinates. 

If I understand correctly, the ramp looks like this:

sin( angle )  =  opposite / hypotenuse

sin( 78° )  =  height / 3        , where height is in meters.

3 sin( 78° )  =  height

height  ≈  2.934

number of dollars you earned  =  (t + 1)(3t - 3)

number of dollars Andrew earned  =  (3t - 5)(t + 2)

number of dollars you earned  =  number of dollars Andrew earned + 2

(t + 1)(3t - 3)  =  (3t - 5)(t + 2) + 2

                                                                  Multiply out  (3t - 5)(t + 2)  and  (t + 1)(3t - 3)

3t² - 3t + 3t - 3  =  3t² + 6t - 5t - 10 + 2

                                                                  Combine like terms.

3t² - 3  =  3t² + t - 8

                                    Add  8  to both sides of the equation.

3t² + 5  =  3t² + t

                                    Subtract  3t²  from both sides of the equation.

5  =  t

Maybe it's a trick question. I was just trying to get you to think.

There isn't a single defined "opposite" of  (8, 5) .

Instead of just saying  " (8, 5)  is the opposite of (-8, 5) "  we need to say something like

(8, 5)  is the opposite over the y-axis of  (-8, 5)

and

(8, 5)  is the opposite over the x-axis of  (8, -5)

It is important to say which axis that the point is "flipped" over.

Nickolas, how would you say that  (8, -5)  and  (8, 5)  are related?

What is the opposite of  (8, 5)  ?

Assuming that     \(f(x)\ =\ 2+\frac{20}{x-3}\)     and     \(g(x)\ =\ -3+\frac{32}{2x+4}\)

We can combine these fractions into one single fraction after getting a common denominator.

\(\mathbf{a.}\qquad\frac{4}{x}-\frac{6}{x}\quad=\quad\frac{4-6}{x}\quad=\quad-\frac{2}{x}\\~\\~\\ \mathbf{b.}\qquad\frac{x}{2}+\frac{2}{3}\quad=\quad\frac{3\,\cdot \,x}{3\,\cdot\,2}+\frac{2\,\cdot\,2}{3\,\cdot\,2}\quad=\quad\frac{3x}{6}+\frac46\quad=\quad\frac{3x+4}{6}\\~\\~\\ \mathbf{c.}\qquad2x\cdot\frac{3}{10}+\frac{x}{10}\quad=\quad\frac{2x\,\cdot\,3}{10}+\frac{x}{10}\quad=\quad\frac{6x}{10}+\frac{x}{10}\quad=\quad\frac{6x+x}{10}\quad=\quad\frac{7x}{10} \)

For this last one, we don't need to get a common denominator because we aren't adding fractions.

We just need to multiply them together and cancel common factors.

\(\mathbf{d.}\qquad\frac{3}{5x^2}\cdot\frac{4x^2}{6x}\quad=\quad\frac{3\,\cdot\,4x^2}{5x^2\,\cdot\,6x}\quad=\quad\frac{3\,\cdot\,2\,\cdot\,2\,\cdot\, x^2}{5\,\cdot\,x^2\,\cdot\,3\,\cdot\,2\cdot\,x}\quad=\quad \frac{{\color{gray}3}\,\cdot\,{\color{gray}2}\,\cdot\,2\,\cdot\, {\color{gray}x^2}}{5\,\cdot\,{\color{gray}x^2}\,\cdot\,{\color{gray}3}\,\cdot\,{\color{gray}2}\cdot\,x}\quad=\quad\frac{2}{5x}\)_

Another word is "reflected."

For example:

If you take  (3, 7)  and make its x-coordinate negative, you will get  (-3, 7) .

Maybe you can imagine that this graph is an open book where the y-axis is the division between two pages.

If you flipped the page, then the two points would meet.

If you took  (3, 7)  and flipped it over the y-axis (like flipping the page), you would get  (-3, 7) .

So  (-3, 7)  is  (3, 7)  flipped over the y-axis.

And you can see that the distance between  (3, 7)  and the y-axis is the same as the distance between  (-3, 7)  and the y-axis.

Does that kind of make sense?