Question

What two numbers multiple to 160, and add up to -4?

Answer 1

Let X be one number and Y be other number. Then, we can write

$X\cdot Y=160$

and

$X-Y=-4$

By moving -Y to the right hand side, we get

$X=Y-4$

By substituting the last expression into the first one, we get

$(Y-4)\cdot Y=160$

which gives

$\begin{gathered} Y^2-4Y=160 \\ Y^2-4Y-160=0 \end{gathered}$

So, we have a quadratic function, which we can solve by means of the quadractic formula:

$Y=\frac{-(-4)\pm\sqrt[]{(-4)^2-4(1)(-160})}{2}$

which gives

$\begin{gathered} Y=\frac{4\pm\sqrt[]{16+640}}{2} \\ Y=\frac{4\pm25.6}{2} \end{gathered}$

So, we have 2 solutions:

$\begin{gathered} Y=\frac{29.6}{2}=14.8 \\ \text{and} \\ Y=\frac{-21.6}{2}=-10.8 \end{gathered}$

By substituting the fist solution into our first equation, we get

$\begin{gathered} X\cdot(14.81)=160 \\ X=\frac{160}{14.8} \\ X=10.81 \end{gathered}$

Now, by substituting the second solution into our first equation, we h ave

$\begin{gathered} X\cdot(-10.8)=160 \\ X=\frac{160}{-10.8} \\ X=-14.81 \end{gathered}$

Lets check one solution. For X=10.81 and Y=14.81 we have

$\begin{gathered} (10.81)(14.8)\approx160 \\ \text{and} \\ 110.81-14.8\approx-4 \end{gathered}$

Now, lets chech the second solution. For X=-14.81 and Y=-10.81 we have

$\begin{gathered} (-10.81)(-14.8)\approx160 \\ \text{and} \\ -14.81+10.81\approx-4 \end{gathered}$

They are correct !! So, one solution is X=10.81 and Y=14.8 and the other solution is X=-14.81 and Y=-10.81.