Chapter 12.1, Problem 66E

To determine

To calculate: The discriminant of the equation $4p^2-3p+2=0$ also describe the nature of roots.

Answer to Problem 66E

The discriminant of the equation $4p^2-3p+2=0$ is $-23$ . The equation has imaginary roots.

Explanation of Solution

Given information:

The equation $4p^2-3p+2=0$ .

Formula used:

The discriminant D of equation $a{x}^2+bx+c=0$ is $D=b^2-4ac$ .

If $D=0$ then equation has real and equal roots.

If $D>0$ then equation has real and unequal roots.

If $D<0$ then equation has imaginary roots.

Calculation:

Consider the equation $4p^2-3p+2=0$ .

The discriminant D of equation $a{x}^2+bx+c=0$ is $D=b^2-4ac$ .

If $D=0$ then equation has real and equal roots.

If $D>0$ then equation has real and unequal roots.

If $D<0$ then equation has imaginary roots.

Apply it,

  $\begin{array}{c}D=b^2-4ac\\ ={\left(-3\right)}^2-4\left(4\right)\left(2\right)\\ =9-32\\ =-23\end{array}$

Since $D<0$ then equation has imaginary roots.

Thus, the discriminant of the equation $4p^2-3p+2=0$ is $-23$ . The equation has imaginary roots.