Chapter 7.3, Problem 7WE

To determine

Find the nature of roots of the equation $z^2+\frac{5}{4}=z$ without solving it.

Answer to Problem 7WE

The roots of the equation are imaginary and conjugate of each other.

Explanation of Solution

Given information: The given quadratic equation is,

  $z^2+\frac{5}{4}=z\mathrm{....}\left(\text{Given}\right)$

Calculation: The given equation $z^2+\frac{5}{4}=z$ can be written in the form $a{x}^2+bx+c=0$ where $\left(a\ne 0\right)$ .

  $z^2-z+\frac{5}{4}=\mathrm{0....}\left(\text{1}\right)$

Now if $\left(D\right)$ is the discriminant of quadratic equation (1) then using the formula $D=b^2-4ac$ ,

  $\begin{array}{c}D={\left(-1\right)}^2-4\left(1\right)\left(\frac{5}{4}\right)\mathrm{....}\left\{\begin{array}{l}a=1,b=-1\\ \text{and }c=\frac{5}{4}\end{array}\right\}\\ =1-5\\ =-\mathrm{4....}\left(\text{which is negative}\right)\end{array}$

Since the value of discriminant $\left(D\right)$ is negative, therefore the roots are imaginary and conjugate of each other.