Chapter 7.3, Problem 8OE

To determine

Find the value of determinant and nature of roots of the equation $k^2-4\sqrt{2}k+4$ . Is the given equation is factorable.

Answer to Problem 8OE

The value of discriminant is 16, the roots are real and unequal and equation is factorable.

Explanation of Solution

Given information: The given quadratic equation is,

  $k^2-4\sqrt{2}k+\mathrm{4....}\left(\text{Given}\right)$

Formula used: The discriminant $\left(D\right)$ of a quadratic equation $a{x}^2+bx+c=0$ where $\left(a\ne 0\right)$ is given as,

  $D=b^2-4ac\mathrm{....}\left(1\right)$

Calculation: The discriminant $\left(D\right)$ for the quadratic equation $k^2-4\sqrt{2}k+4$ is evaluated as,

  $\begin{array}{c}D={\left(-4\sqrt{2}\right)}^2-4\left(1\right)\left(4\right)\mathrm{....}\left\{\begin{array}{l}a=1,b=-4\sqrt{2}\\ \text{and }c=4\end{array}\right\}\\ =32-16\\ =16\end{array}$

Since the value of discriminant is positive and a perfect square value, therefore the two roots are real and unequal.

Also the perfect square value of discriminant states that the equation is factorable.