Chapter 7.3, Problem 11WE

To determine

Find the nature of roots of the equation $2u^2-5u+\sqrt{8}=0$ without solving it.

Answer to Problem 11WE

The equation has two real and unequal roots.

Explanation of Solution

Given information: The given quadratic equation is,

  $2u^2-5u+\sqrt{8}=\mathrm{0....}\left(\text{Given}\right)$

Calculation: The given equation can be written as,

  $2u^2-5u+2\sqrt{2}=\mathrm{0....}\left(\text{1}\right)$

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

  $\begin{array}{c}D={\left(-5\right)}^2-4\left(2\right)\left(2\sqrt{2}\right)\mathrm{....}\left\{\begin{array}{l}a=2,b=-5\\ \text{and }c=2\sqrt{2}\end{array}\right\}\\ =25-16\sqrt{2}\\ =\mathrm{2.372....}\left(\text{Approx}\text{.}\right)\\ >0\end{array}$

Since the value of discriminant $\left(D\right)$ is $2.372$ which is greater than zero, therefore the roots are real and unequal to each other.