Chapter 4.7, Problem 5LC

Solution

To find: The discriminant of the equation $-x^2+2x-9=0$ and the number of real solutions.

The discriminant of the equation is $-32$ and the equation has no real solution.

Given information:

The equation $-x^2+2x-9=0$ .

Concept used:

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

If $b^2-4ac<0$ , the equation has no real solutions.

If $b^2-4ac=0$ , the equation has one real solution.

If $b^2-4ac>0$ , the equation has two real solutions.

Calculation:

Identify the values of $a$ , $b$ and $c$ for the equation $-x^2+2x-9=0$ as:

  $a=-1$ , $b=2$ and $c=-9$ .

Now, substitute the values of $a$ , $b$ and $c$ in the expression $b^2-4ac$ to find the discriminant.

  $\begin{array}{c}{b}^2-4ac=2^2-4\left(-1\right)\left(-9\right)\\ =4-36\\ =-32\end{array}$

So, the discriminant is $-32$ .

Since the discriminant is negative, the equation has no real solutions.

Conclusion:

The discriminant of the equation is $-32$ and the equation has no real solution.