+2446 The maximum or minimum of any quadratic is always at the input value of \(\frac{-b}{2a}\) where the standard form of a quadratic is \(ax^2+bx+c=0\) .
| \(a=-4, b=16;\\ x=\frac{-b}{2a}\) | Substitute in the known values for a and b . |
| \(x=\frac{-16}{2*-4}\) | Solve for x by simplifying completely. |
| \(x=2\) | This is the optimal number of items sold, in thousands, in order to make maximum profit. Let's substitute that into the original function to determine the profit with that number of items sold. |
| \(x=2;\\ p(2)=-4*2^2+16*2-7\) | Yet again, just simplify from here. |
| \(p(2)=-4*4+32-7\) | |
| \(p(2)=-16+25\) | |
| \(p(2)=9\) | This is the maximum profit that this business, in thousands of dollars, makes. |
+2446 Hi, nellycrane!
I think you will be better off if I give you an easier problem like \(x^2-4 \) . You are probably well aware that this is a difference of squares. \((x+2)(x-2)\) would be the corresponding factorization.
Now, let's say that you want to factor \(x^2-30\) . This is not possible if you restrict yourself to the rational number set. However, it can be factored as \((x+\sqrt{30})(x-\sqrt{30})\).
In general, \(a^2-b^2=\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)\) . You can apply this knowledge to the problems at hand.
\(x^2+50\\ a=x^2,b=-50;\\ (\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})\\ (\sqrt{x^2}+\sqrt{-50})(\sqrt{x^2}-\sqrt{-50}) \)
You can simplify this to get the factorization amongst the complex numbers. Good luck!
+2446 This problem is easier than it looks at first glance. If you utilize clever algebraic manipulation, this problem becomes simpler.
| \(x-y=16\) | \(xy=23\) |
| \((x-y)^2=16^2\) | |
| \(\boxed{1}\hspace{1mm}x^2-2xy+y^2=256\) | \(\boxed{2}\hspace{1mm}2xy=46\) |
Notice what I have done. I have manipulated the information I know about these real numbers, x and y , and I am manipulating it in a way that is much more convenient for this particular problem. The only thing left to do is add the equations together.
\(\boxed{1}\hspace{1mm}x^2-2xy+y^2=256\\ \boxed{2}\hspace{5mm}+2xy\hspace{10mm}=46\\ \overline{\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad}\\ \hspace{18mm}x^2+y^2=302\)
