Chapter 7, Problem 4CT

To determine

To find: the numbers.

Answer to Problem 4CT

The numbers are $\frac{7\pm \sqrt{5}}{2},\frac{22}{7\pm \sqrt{5}}$ .

Explanation of Solution

Given information:

Sum of numbers $=7$ and product of numbers $=11$ .

Calculation: to find the numbers,

Let $a,b$ represents the numbers,

Since, Sum of numbers $=7$ and product of numbers $=11$

$\begin{array}{l}a+b=7\\ ab=11\end{array}$

Now, simplify the equations

$\begin{array}{l}a+b=7\text{ [eq}\text{. 1]}\\ ab=11\\ b=\frac{11}{a}\text{ [eq}\text{. 2]}\\ a+\frac{11}{a}=7\text{ [substitute from eq}\text{. 2]}\\ a^2+11=7a\text{ [simplify]}\\ a^2-7a+11=0\text{ [subtract]}\\ a=\frac{-\left(-7\right)\pm \sqrt{{\left(-7\right)}^2-4\left(1\right)\left(11\right)}}{2\left(1\right)}\text{ [use the quadratic equation]}\\ a=\frac{7\pm \sqrt{49-44}}{2}\text{ [simplify]}\\ a=\frac{7+\sqrt{5}}{2},\frac{7-\sqrt{5}}{2}\text{ [subtract]}\\ \left(\frac{7\pm \sqrt{5}}{2}\right)b=11\text{ [substitute]}\\ b=\frac{22}{7+\sqrt{5}},\frac{22}{7-\sqrt{5}}\text{ [cross multiplication]}\end{array}$

Thus, the numbers are $\frac{7\pm \sqrt{5}}{2},\frac{22}{7\pm \sqrt{5}}$ .