Chapter 7.3, Problem 8WE

To determine

Find the nature of roots of the equation $\frac{r^2}{4}+1=r$ without solving it.

Answer to Problem 8WE

The equation has double root.

Explanation of Solution

Given information: The given quadratic equation is,

  $\frac{r^2}{4}+1=r\mathrm{....}\left(\text{Given}\right)$

Calculation: The given equation $\frac{r^2}{4}+1=r$ can be written in the form $a{x}^2+bx+c=0$ where $\left(a\ne 0\right)$ .

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

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

  $\begin{array}{c}D={\left(-1\right)}^2-4\left(\frac{1}{4}\right)\left(1\right)\mathrm{....}\left\{\begin{array}{l}a=\frac{1}{4},b=-1\\ \text{and }c=1\end{array}\right\}\\ =1-1\\ =\mathrm{0....}\left(\text{which is 0}\right)\end{array}$

Since the value of discriminant $\left(D\right)$ is zero, therefore the roots are real and equal to each other. In other words the equation has double root.