Chapter 3.1, Problem 73E

a.

To determine

To write the given expression in given form.

Answer to Problem 73E

  $a^2=mn,xm+xn=0$

Explanation of Solution

Given information :

Given expression is $x^2+a^2$ .

To write the given expression in the form $\left(x+m\right)\left(x+n\right)$ we shall compare the coefficients.

Hence we get,

  $\begin{array}{l}{x}^2+a^2=\left(x+m\right)\left(x+n\right)\\ x^2+a^2=x^2+xn+xm+mn\end{array}$

Now we compare coefficients on both sides we get the two necessary conditions as,

  $a^2=mn,xm+xn=0$

b.

To determine

To solve equation for m and n.

Answer to Problem 73E

m and n are complex numbers.

Explanation of Solution

Two conditions we determined were,

  $a^2=mn,xm+xn=0$

Now we know,

  $\begin{array}{l}xm+xn=0\\ x\left(m+n\right)=0\\ m+n=0\\ m=-n\end{array}$

Now we shall put $m=-n$ in equation $a^2=mn$

We get,

  $\begin{array}{l}-n^2=a^2\\ n=\sqrt{-a}\end{array}$

Therefore n is a complex number.