Chapter 7.3, Problem 9WE

To determine

Find the nature of roots of the equation $d^2+\frac{7}{3}d=2$ without solving it.

Answer to Problem 9WE

The equation has two real and unequal roots.

Explanation of Solution

Given information: The given quadratic equation is,

  $d^2+\frac{7}{3}d=\mathrm{2....}\left(\text{Given}\right)$

Calculation: The given equation $d^2+\frac{7}{3}d=2$ can be written in the form $a{x}^2+bx+c=0$ where $\left(a\ne 0\right)$ .

  $d^2+\frac{7}{3}d-2=\mathrm{0....}\left(\text{1}\right)$

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

  $\begin{array}{c}D={\left(\frac{7}{3}\right)}^2-4\left(1\right)\left(-2\right)\mathrm{....}\left\{\begin{array}{l}a=1,b=\frac{7}{3}\\ \text{and }c=-2\end{array}\right\}\\ =\frac{49}{9}+8\\ =\frac{121}{9}\\ >0\end{array}$

Since the value of discriminant $\left(D\right)$ is $\frac{121}{9}$ which is greater than zero, therefore the roots are real and unequal to each other.