Answer
The two numbers are $5+i\sqrt{15}$ and $5-i\sqrt{15}$.
Work Step by Step
Consider the two numbers $x$ and $y$ whose sum is $10$ and product is $40$.
The sum of the two numbers is,
$x+y=10$
The product of the numbers is,
$xy=40$
Divide $x$ on both sides.
$\begin{align}
& \frac{xy}{x}=\frac{40}{x} \\
& y=\frac{40}{x}
\end{align}$
Substitute $y=\frac{40}{x}$ in the equation$x+y=10$.
$\begin{align}
& x+\frac{40}{x}=10 \\
& \frac{{{x}^{2}}+40}{x}=10 \\
& {{x}^{2}}+40=10x \\
& {{x}^{2}}-10x+40=0
\end{align}$
Compare the equation ${{x}^{2}}-10x+40=0$ with $a{{x}^{2}}+bx+c$.
$\begin{align}
& a=1 \\
& b=-10 \\
& c=40
\end{align}$
Substitute $a=1$, $b=-10$ and $c=40$ in the formula $x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.
$\begin{align}
& x=\frac{-\left( -10 \right)\pm \sqrt{{{\left( -10 \right)}^{2}}-4\left( 1 \right)\left( 40 \right)}}{2\left( 1 \right)} \\
& =\frac{10\pm \sqrt{100-160}}{2} \\
& =\frac{10\pm \sqrt{-60}}{2}
\end{align}$
Use the property $\sqrt{-b}=i\sqrt{b}$.
\[\begin{align}
& x=\frac{10\pm i\sqrt{60}}{2} \\
& =\frac{10\pm i\sqrt{4\cdot 15}}{2} \\
& =\frac{10\pm 2i\sqrt{15}}{2} \\
& =5\pm i\sqrt{15}
\end{align}\]
So, the two numbers are $5+i\sqrt{15}$ and $5-i\sqrt{15}$.
Check that the sum of the two complex numbers $5+i\sqrt{15}$ and $5-i\sqrt{15}$ is $10$.
$\begin{align}
& \left( 5+i\sqrt{15} \right)+\left( 5-i\sqrt{15} \right)=\left( 5+5 \right)+\left( \sqrt{15}-\sqrt{15} \right)i \\
& =10+\left( 0 \right)i \\
& =10
\end{align}$
So, it shows that the complex numbers $5+i\sqrt{15}$ and $5-i\sqrt{15}$ satisfy the condition.
Check that the product of two complex numbers $5+i\sqrt{15}$ and $5-i\sqrt{15}$ is $40$.
Use the FOIL method.
$\begin{align}
& \left( 5+i\sqrt{15} \right)\left( 5-i\sqrt{15} \right)=25-5i\sqrt{15}+5i\sqrt{15}-{{\left( \sqrt{15} \right)}^{2}}{{i}^{2}} \\
& =25-15{{i}^{2}}
\end{align}$
Replace the value ${{i}^{2}}=-1$.
$\begin{align}
& \left( 5+i\sqrt{15} \right)\left( 5-i\sqrt{15} \right)=25-15\left( -1 \right) \\
& =25+15 \\
& =40
\end{align}$
So, it shows that the complex numbers $5+i\sqrt{15}$ and $5-i\sqrt{15}$ satisfy the condition.
Therefore, the two numbers are $5+i\sqrt{15}$ and $5-i\sqrt{15}$.
