CPhill

y2 + 6xy + 9x2      This is a perfect square trinomial....the key to factoring these is to take the square root of the first and last terms and separate them by the sign on the middle term....then just square the whole thing...

So we have.....

(y + 3x)

Now...just square the whole thing

(y + 3x)²

And there you are  !!!

Here's my take on this one :

Melody is correct

The amplitude is 1

The vertical shift is "up' 3 units

The horizontal shift is to the left by 5 rads

Note that the "4" tells us how many periods there are in 2pi...thus, the normal cosine graph is 'compressed" by a factor of 4 !!!

18 - 24x + 8x^2   Let's start by taking out the greatest common factor (2)    .....so we have....

2(9 - 12x + 4x^2)   And the polynomial in the parentheses is a perfect square trinomial, so we have.....

2(3 - 2x) (3 - 2x)  =   2(3 - 2x)²

And that's it !!

Yep....welcome back, Aziz !!!  We've been missing you !!!

Very nice proof, Melody !!! 

Thanks, Melody.......I see now what the questioner meant !!!!......disregard my answer !!!

BTW.....I liked the way you did that one....very clever......I haven't seen that technique before, but I intend to add it to my "tool chest".....!!!!!

Good answer, Haley  !!!

$${\mathtt{69}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{4}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{3}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{62}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{7}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{7}}}{{\mathtt{23}}}}\\
{\mathtt{x}} = {\frac{{\mathtt{1}}}{{\mathtt{3}}}}\\
{\mathtt{x}} = -{\mathtt{1}}{i}\\
{\mathtt{x}} = {i}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{0.304\: \!347\: \!826\: \!086\: \!956\: \!5}}\\
{\mathtt{x}} = {\mathtt{0.333\: \!333\: \!333\: \!333\: \!333\: \!3}}\\
{\mathtt{x}} = -{\mathtt{1}}{i}\\
{\mathtt{x}} = {i}\\
\end{array} \right\}$$

So, this polynomial has two "real" and two "non-real" roots  !!!

.72 is a non-repeating decimal whose fractional equivalent is just 72/100

m³ = 512    take the cube root of both sides

m= ³√512 = 8