How to solve this simple equation $\frac{46}{y} + y = 25$? How do I solve this simple equation?  $\frac{46}{y} + y = 25$
I know that the answer is $2$, but how do I arrive at that? 
 A: $$\frac{46}{y}+y = 25$$
Here, assuming $y \neq 0$, you can multiply both sides of the equation by $y$, yielding
$$46+y^2 = 25y$$
$$y^2-25y+46 = 0$$
Here, you can solve the quadratic equation  by factoring, completing the square, or the Quadratic Formula, but the first way is the simplest, considering $a = 1$:
$$(y+y_1)(y+y_2) = 0$$
The key here is to ask yourself: which two numbers multiply to give $+46$ and add to give $-25$? Clearly the two numbers must be negative, and you can figure out they’re $-23$ and $-2$. So, the factored equation becomes
$$(y-23)(y-2) = 0$$
Setting either factor equal to $0$ yields

*

*$$y-23 = 0 \iff y = 23$$

*$$y-2 = 0 \iff y = 2$$
You could also use the Quadratic Formula if desired:

$$ax^2+bx+c = 0 \iff x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
$$y = \frac{-(-25)\pm\sqrt{(-25)^2-4(1)(46)}}{2(1)} = \frac{25\pm\sqrt{441}}{2} = \frac{25\pm 21}{2}$$
$$y = 23; \quad y = 2$$

A: $$y^2-25y+46=0\implies y=2,23$$
you can use middle term method or the quadratic formula.
the formula is...
if $$ax^2+bx+c=0,then~~~ x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$
A: Since you see $\frac{46}{y}$ in there, it's a good idea to multiply both sides by $y$:
\begin{align*}
\left(\frac{46}{y} + y\right) \cdot y &= 25 \cdot y \\
46 + y^2 &= 25y
\end{align*}
Now we bring all the terms over to one side:
$$
y^2 - 25y + 46 = 0
$$
so you have a quadratic equation, and you can solve it. (Quadratic formula or factoring!)
If you happen to get $y = 0$ as a solution to the equation, you have to throw it out, since at the beginning there was $\frac{46}{y}$, which means that $y$ was not $0$. But here it turns out that $y = 0$ is not a solution.
A: Multiply both sides by $y$ and you get
$$46 + y^2 = 25y$$
$$\Leftrightarrow 0 = y^2 -25y + 46$$
Now calculating the discriminant $D = (-25)^2 -4\cdot 46 = 625-184 = 441 = 21^2$ gives
$$y = \frac{25 \pm \sqrt{21^2}}{2}$$
So $y = \frac{25+21}{2} = 23$ or $y = \frac{25-21}{2} = 2$.
