Chapter 4.7, Problem 27PPSE

Solution

To find: The discriminant of the equation $-4x^2+20x-25=0$ and the number of real solutions.

The discriminant of the equation is $0$ and the equation has one real solution.

Given information:

The equation $-4x^2+20x-25=0$ .

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 $-4x^2+20x-25=0$ as:

  $a=-4$ , $b=20$ and $c=-25$ .

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

  $\begin{array}{c}{b}^2-4ac={20}^2-4\left(-4\right)\left(-25\right)\\ =400-4\left(-4\right)\left(-25\right)\\ =400-400\\ =0\end{array}$

So, the discriminant is $0$ .

Since the discriminant is zero, the equation has one real solution.

Conclusion:

The discriminant of the equation is $0$ and the equation has one real solution.