Answer
$\text{Discriminant: }
0
\\\text{Number of solutions: }
1
\\\text{Type of solutions:
rational solution
}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the discriminant of the given equation, $
25x^2+110x+121=0
,$ identify first the values of $a,b,$ and $c$ in the quadratic expression $ax^2+bx+c.$ Then use the Discriminant Formula. If the value of the discriminant is less than zero, then there are $\text{
2 nonreal complex solutions
.}$ If the value is $0,$ then there is $\text{
1 distinct rational solution
.}$ If the value of the discriminant is a positive perfect square, then there are $\text{
2 rational solutions
.}$ Finally, if the value of the discriminant is positive but not a perfect square, there are $\text{
2 irrational solutions
.}$
$\bf{\text{Solution Details:}}$
In the given equation above, $a=
25
,$ $b=
110
,$ and $c=
121
.$ Using the Discriminant Formula which is given by $b^2-4ac,$ the value of the discriminant is
\begin{array}{l}\require{cancel}
(110)^2-4(25)(121)
\\\\=
12100-12100
\\\\=
0
.\end{array}
Since the discriminant is $\text{
is equal to zero
,}$ then there is $\text{
1 rational solutions
.}$
Hence, the given equation has the following properties:
\begin{array}{l}\require{cancel}
\text{Discriminant: }
0
\\\text{Number of solutions: }
1
\\\text{Type of solutions:
rational solution
}
.\end{array}
