Chapter 12.4, Problem 8WE

To determine

To write: the value of the discriminant of given equation and to find the number of different real numbers roots in the equation.

Answer to Problem 8WE

The value of the discriminant is $72$ .

The number of different real roots is two.

Explanation of Solution

Given information:

Given equation,

  $3a^2-2a-6=0$

Calculation:

Consider the given equation:

  $3a^2-2a-6=0$

The discriminant of quadratic equation of the form $a^2+bx+c=0$ is given by:

  $D=b^2-4ac$ .

Now, nature of roots can be determined by the following:

1. $D>0$, the equation has two different real number roots.
2. $D=0$, the equation has two equal real roots.
3. $D<0$, the equation has two complex conjugate roots.

  $a=3,b=-2$ and $c=-6$

Now, evaluate the discriminant as shown:

  $\begin{array}{c}D=b^2-4ac\\ ={\left(-2\right)}^2-4\left(3\right)\left(-6\right)\\ =4+72\\ =76\end{array}$

So, the value of the discriminant is $72$ .

Since D is positive.

  $\therefore$ The number of different real roots is two.