CPhill

Yeah....those internet searches really pay off, sometimes.....(LOL!!!)

Because it has a lot of "problems"

This is similar to the problem I just worked....here's the graph here.....the "x" value is the part reserved for the triangle = about 5.7 m  and the part reserved for the square = (10 - 5.7) = about 4.3 m

Let 22-x be the length reserved for the square and let (x) be the length reserved for the triangle.

So (22-x)/4 is be length of the side of the square and (x/3) is the length of the side of the equilateral triangle.

So the area of the square is given by  [(22-x)/4]^2 = (1/16)[22-x]^2. And the area of the triangle is given by

(1/2)(x/3)^2sin(60) = (1/2)(1/9)x^2(√3/2) = (√3/36)x^2

So the total area is given by

A = (1/16)[22-x]^2 + (√3/36)x^2

Every even number can be written in the form  2n....Therefore..

2n * 2n  = 4n^2

But 4n^2   =  2(2n^2) and this is divisible by 2 for any integer n ....and divsibility by 2 is the definition of an even number......so an even times an even is always even.

Note that we're just subbing "3a" in for "x" .....so we have.......

f(3a) = 3a + 2 

This one is pretty easy.....

We have

sin(60) = opp/ hyp = opp/ 12

So the opposite side - how high the ladder will reach  -  is just   (12*√3) /2 = 6√3 ft. = about 10.4 ft.

Thanks for that resource, Melody.....I saw this question earlier, but I didn't know how to proceed with a good explanation.....that website looks to be pretty informative......multiplying "by hand" seems to be a lost art !!!!

109 is a prime number, so this can't be factored in a real number form!!!!

Also, "factoring" the lead term - x^3 - would involve x and x^2. So we would have an additional "x" term that didn't exist in the original problem......

Thus, this one is irreducible to a factored form.......

This doesn't have any real solutions....

Here are the non-real ones

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{19}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{2}}{\mathtt{\,-\,}}{\sqrt{{\mathtt{15}}}}{\mathtt{\,\times\,}}{i}\\
{\mathtt{x}} = {\sqrt{{\mathtt{15}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{2}}{\mathtt{\,-\,}}{\mathtt{3.872\: \!983\: \!346\: \!207\: \!416\: \!9}}{i}\\
{\mathtt{x}} = {\mathtt{2}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3.872\: \!983\: \!346\: \!207\: \!416\: \!9}}{i}\\
\end{array} \right\}$$