Question

The sum of two numbers is $22$and the product is $-240$.Find the numbers.

Short answer

The solution are Line and Hyperbola.

Step-by-step solution

Step 1. Given Information

Given that the two numbers is $22$and product is$240$.

Step 2. Standard form

The equation can be denoted as,

$$\begin{gathered} \mathrm{x}+\mathrm{y}=22\mathrm{is}\mathrm{linear} \\ \mathrm{xy}=-240\mathrm{is}\mathrm{non}-\mathrm{linear} \end{gathered}$$

That is hyperbola equation.

Step 3. Standard equation

It can be denoted as,

$$\begin{gathered} \mathrm{x}+\mathrm{y}-22=0 \\ \mathrm{xy}=-240 \\ \mathrm{x}=\frac{-240}{\mathrm{x}} \end{gathered}$$

The standard form of$\mathrm{x}+\mathrm{y}=22\mathrm{is}\mathrm{x}+\mathrm{y}-22=0\mathrm{and}\mathrm{xy}=-240\mathrm{is}\mathrm{x}=\frac{-240}{\mathrm{x}}.$

Step 4. Number of equation

Let the first and second number are x and y. On solving the equation,

$$\begin{gathered} \mathrm{x}+\mathrm{y}=22 \\ \mathrm{y}=22-\mathrm{x} \\ \mathrm{xy}=-240 \\ \mathrm{x}(22-\mathrm{x})=-240 \\ 22\mathrm{x}-{\mathrm{x}}^2=-240 \\ {\mathrm{x}}^2-22-240=0 \\ \mathrm{a}=1,\mathrm{b}=-22,\mathrm{c}=-240 \\ \mathrm{x}=\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}} \\ \mathrm{x}=\frac{22\pm \sqrt{{(-22)}^2-4\times 1\times -240}}{2\times 1} \\ \mathrm{x}=-8,30 \end{gathered}$$

Step 5. Graph of the equation

The graph of linear and non linear equation can be drawn as,

Step 6. Intersection point

The hyperbola and line can be intersected at $(-8,30)\mathrm{and}(30,-8).$It can be verified as,

$$\begin{gathered} \mathrm{x}+\mathrm{y}=22 \\ -8+30=22 \\ 22=22 \\ \mathrm{xy}=-240 \\ \left(30\right)(-8)=-240 \\ -240=-240 \end{gathered}$$

The point of intersection satisfied both the equation.