Chapter 4, Problem 23CT

Solution

To find: The discriminant of the equation $-4x^2-4x+1=0$ and the number of real and imaginary solutions.

The discriminant of the equation is $32$ and, the equation has two real solutions and zero imaginary solution.

Given information:

The equation $-4x^2-4x+1=0$ .

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 $-4x^2-4x+1=0$ as:

  $a=-4$ , $b=-4$ and $c=1$ .

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

  $\begin{array}{c}{b}^2-4ac={\left(-4\right)}^2-4\left(-4\right)\cdot 1\\ =16-\left(-16\right)\\ =32\end{array}$

So, the discriminant is $32$ .

Since the discriminant is positive, the equation has two real solutions.

As the quadratic equation can at most two solutions, the equation has zero imaginary solution.

Conclusion:

The discriminant of the equation is $32$ and, the equation has two real solutions and zero imaginary solution.